Topics in Topological Graph Theory

The use of topological ideas to explore various aspects of graph theory, and vice versa, is a fruitful area of research. There are links with other areas of mathematics, such as design theory and geometry, and increasingly with such areas as computer networks where symmetry is an important feature. Other books cover portions of the material here, but there are no other books with such a wide scope. This book contains fifteen expository chapters written by acknowledged international experts in the field. Their well-written contributions have been carefully edited to enhance readability and to standardize the chapter structure, terminology and notation throughout the book. To help the reader, there is an extensive introductory chapter that covers the basic background material in graph theory and the topology of surfaces. Each chapter concludes with an extensive list of references. lowell w. beineke is Schrey Professor of Mathematics at Indiana University–Purdue University Fort Wayne, where he has been since receiving his Ph.D. from the University of Michigan under the guidance of Frank Harary. His graph theory interests are broad, and include topological graph theory, line graphs, tournaments, decompositions and vulnerability. With Robin Wilson he edited Selected Topics in Graph Theory (3 volumes), Applications of Graph Theory, Graph Connections and Topics in Algebraic Graph Theory. Until recently he was editor of the College Mathematics Journal. robin j. wilson is Professor of Pure Mathematics at The Open University, UK, and Emeritus Professor of Geometry at Gresham College, London. After graduating from Oxford, he received his Ph.D. in number theory from the University of Pennsylvania. He has written and edited many books on graph theory and the history of mathematics, including Introduction to Graph Theory and Four Colours Suffice, and his research interests include graph colourings and the history of combinatorics. He has won a Lester Ford Award and a George Polya´ Award from the MAA for his expository writing. jonathan l. gross, Professor of Computer Science at Columbia University, served as an academic consultant for this volume. His mathematical work in topology and graph theory have earned him an Alfred P. Sloan Fellowship, an IBM Postdoctoral Fellowship, and numerous research grants. With Thomas Tucker, he wrote Topological Graph Theory and several fundamental pioneering papers on voltage graphs and on enumerative methods. He has written and edited eight books on graph theory and combinatorics, seven books on computer programming topics, and one book on cultural sociometry. thomas w. tucker, Charles Hetherington Professor of Mathematics at Colgate University, also served as an academic consultant for this volume. He has been at Colgate University since 1973, after a Ph.D. in 3-manifolds from Dartmouth in 1971 and a post-doctoral position at Princeton. He is co-author (with Jonathan Gross) of Topological Graph Theory. His early publications were on non-compact 3-manifolds, then topological graph theory, but his recent work is mostly algebraic, especially distinguishability and the group-theoretic structure of symmetric maps.

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit http://www.cambridge.org/uk/series/sSeries.asp?code=EOM

68 R. Goodman and N. R. Wallach Representations and Invariants of the Classical Groups 69 T. Beth, D. Jungnickel, and H. Lenz Design Theory I, 2nd edn 70 A. Pietsch and J. Wenzel Orthonormal Systems for Banach Space Geometry 71 G. E. Andrews, R. Askey and R. Roy Special Functions 72 R. Ticciati Quantum Field Theory for Mathematicians 73 M. Stern Semimodular Lattices 74 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations I 75 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations II 76 A. A. Ivanov Geometry of Sporadic Groups I 77 A. Schinzel Polynomials with Special Regard to Reducibility 78 T. Beth, D. Jungnickel and H. Lenz Design Theory II, 2nd edn 79 T. W. Palmer Banach Algebras and the General Theory of *-Albegras II 80 O. Stormark Lie’s Structural Approach to PDE Systems 81 C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables 82 J. P. Mayberry The Foundations of Mathematics in the Theory of Sets 83 C. Foias, O. Manley, R. Rosa and R. Temam Navier–Stokes Equations and Turbulence 84 B. Polster and G. Steinke Geometries on Surfaces 85 R. B. Paris and D. Kaminski Asymptotics and Mellin–Barnes Integrals 86 R. McEliece The Theory of Information and Coding, 2nd edn 87 B. A. Magurn An Algebraic Introduction to K-Theory 88 T. Mora Solving Polynomial Equation Systems I 89 K. Bichteler Stochastic Integration with Jumps 90 M. Lothaire Algebraic Combinatorics on Words 91 A. A. Ivanov and S. V. Shpectorov Geometry of Sporadic Groups II 92 P. McMullen and E. Schulte Abstract Regular Polytopes 93 G. Gierz et al. Continuous Lattices and Domains 94 S. R. Finch Mathematical Constants 95 Y. Jabri The Mountain Pass Theorem 96 G. Gasper and M. Rahman Basic Hypergeometric Series, 2nd edn 97 M. C. Pedicchio and W. Tholen (eds.) Categorical Foundations 98 M. E. H. Ismail Classical and Quantum Orthogonal Polynomials in One Variable 99 T. Mora Solving Polynomial Equation Systems II 100 E. Olivieri and M. Eulalia´ Vares Large Deviations and Metastability 101 A. Kushner, V. Lychagin and V. Rubtsov Contact Geometry and Nonlinear Differential Equations 102 L. W. Beineke and R. J. Wilson (eds.) with P. J. Cameron Topics in Algebraic Graph Theory 103 O. Staffans Well-Posed Linear Systems 104 J. M. Lewis, S. Lakshmivarahan and S. K. Dhall Dynamic Data Assimilation 105 M. Lothaire Applied Combinatorics on Words 106 A. Markoe Analytic Tomography 107 P. A. Martin Multiple Scattering 108 R. A. Brualdi Combinatorial Matrix Classes 110 M.-J. Lai and L. L. Schumaker Spline Functions on Triangulations 111 R. T. Curtis Symmetric Generation of Groups 112 H. Salzmann, T. Grundhofer,¨ H. Hahl¨ and R. Lowen¨ The Classical Fields 113 S. Peszat and J. Zabczyk Stochastic Partial Differential Equations with Levy´ Noise 114 J. Beck Combinatorial Games 116 D. Z. Arov and H. Dym J-Contractive Matrix Valued Functions and Related Topics 117 R. Glowinski, J.-L. Lions and J. He Exact and Approximate Controllability for Distributed Parameter Systems 118 A. A. Borovkov and K. A. Borovkov Asymptotic Analysis of Random Walks 119 M. Deza and M. Dutour Sikiric´ Geometry of Chemical Graphs 120 T. Nishiura Absolute Measurable Spaces 121 M. Prest Purity, Spectra and Localisation 122 S. Khrushchev Orthogonal Polynomials and Continued Fractions: From Euler’s Point of View 123 H. Nagamochi and T. Ibaraki Algorithmic Aspects of Graph Connectivity 124 F. W. King Hilbert Transforms I 125 F. W. King Hilbert Transforms II 126 O. Calin and D.-C. Chang Sub-Riemannian Geometry 127 M. Grabisch, J.-L. Marichal, R. Mesiar and E. Pap Aggregation Functions Leonhard Euler (1707–1783), the founder of topological graph theory. Topics in Topological Graph Theory

Edited by

LOWELL W. BEINEKE Indiana University–Purdue University Fort Wayne

ROBIN J. WILSON The Open University

Academic Consultants

JONATHAN L. GROSS Columbia University

THOMAS W. TUCKER Colgate University cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao˜ Paulo, Delhi

Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org Information on this title: www.cambridge.org/9780521802307

c Cambridge University Press 2009

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2009

Printed in the United Kingdom at the University Press, Cambridge

A catalogue record for this publication is available from the British Library

ISBN 978-0-521-80230-7 hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. This book is dedicated to the memory of Gerhard Ringel (1919–2008), one of the pioneers of modern topological graph theory.

Contents

Foreword by Jonathan L. Gross and Thomas W. Tucker page xv Preface xvii

Introduction1 LOWELL W. BEINEKE and ROBIN J. WILSON 1. Graph theory 1 2. Graphs in the plane 10 3. Surfaces 12 4. Graphs on surfaces 14

1 Embedding graphs on surfaces 18 JONATHAN L. GROSS and THOMAS W. TUCKER 1. Introduction 18 2. Graphs and surfaces 19 3. Embeddings 20 4. Rotation systems 23 5. Covering spaces and voltage graphs 26 6. Enumeration 29 7. Algorithms 30 8. Graph minors 31

2 Maximum genus 34 JIANER CHEN and YUANQIU HUANG 1. Introduction 34 2. Characterizations and complexity 36 3. Kuratowski-type theorems 38 4. Upper-embeddability 39 5. Lower bounds 40

ix x Contents

3 Distribution of embeddings 45 JONATHAN L. GROSS 1. Introduction 45 2. Enumerating embeddings by surface type 48 3. Total embedding distributions 51 4. Congruence classes 53 5. The unimodality problem 55 6. Average genus 56 7. Stratiﬁcation of embeddings 59

4 Algorithms and obstructions for embeddings 62 BOJAN MOHAR 1. Introduction 62 2. Planarity 64 3. Outerplanarity and face covers 66 4. Disc embeddings and the 2-path problem 68 5. Graph minors and obstructions 69 6. Algorithms for embeddability in general surfaces 73 7. Computing the genus 75

5 Graph minors: generalizing Kuratowski’s theorem 81 R. BRUCE RICHTER 1. Introduction 81 2. Graph decompositions 84 3. Linked decompositions 88 4. Graphs with bounded tree-width 94 5. Finding large grids 99 6. Embedding large grids 107

6 Colouring graphs on surfaces 111 JOAN P. HUTCHINSON 1. Introduction 111 2. High-end colouring 113 3. A transition from high-end to low-end colouring 116 4. Colouring graphs with few colours 119 5. Girth and chromatic number 124 6. List-colouring graphs 125 7. More colouring extensions 127 8. An open problem 129 Contents xi

7 Crossing numbers 133 R. BRUCE RICHTER and G. SALAZAR 1. Introduction 133 2. What is the crossing number? 135 3. General bounds 137 4. Applications to geometry 139 5. Crossing-critical graphs 139 6. Other families of graphs 143 7. Algorithmic questions 144 8. Drawings in other surfaces 146 9. Conclusion 147

8 Representing graphs and maps 151 TOMAZˇ PISANSKI and ARJANA ZITNIKˇ 1. Introduction 151 2. Representations of graphs 152 3. Energy and optimal representations 155 4. Representations of maps 163 5. Representations of maps in the plane 170 6. Representations of incidence geometries and related topics 174

9 Enumerating coverings 181 JIN HO KWAK and JAEUN LEE 1. Introduction 181 2. Graph coverings 183 3. Regular coverings 185 4. Surface branched coverings 190 5. Regular surface branched coverings 193 6. Distribution of surface branched coverings 195 7. Further remarks 196

10 Symmetric maps 199 JOZEF SIRˇ A´ Nˇ and THOMAS W. TUCKER 1. Introduction 199 2. Representing maps algebraically 200 3. Regular maps 205 4. Cayley maps 210 5. Regular Cayley maps 212 6. Edge-transitive maps 218 7. Maps and mathematics 221 xii Contents

11 The genus of a group 225 THOMAS W. TUCKER 1. Introduction 225 2. Symmetric embeddings and groups acting on surfaces 226 3. Quotient embeddings and voltage graphs 228 4. Inequalities 232 5. Groups of low genus 235 6. Genera of families of groups 239

12 Embeddings and geometries 245 ARTHUR T. WHITE 1. Introduction 245 2. Surface models 248 3. Projective geometries 250 4. Afﬁne geometries 253 5. 3-conﬁgurations 256 6. Partial geometries 260 7. Regular embeddings for PG(2,n) 264 8. Problems 265

13 Embeddings and designs 268 M. J. GRANNELL and T. S. GRIGGS 1. Introduction 268 2. Steiner triple systems and triangulations 270 3. Recursive constructions 273 4. Small systems 278 5. Cyclic embeddings 280 6. Concluding remarks 284

14 Inﬁnite graphs and planar maps 289 MARK E. WATKINS 1. Introduction 289 2. Ends 290 3. Automorphisms 293 4. Connectivities 295 5. Growth 300 6. Inﬁnite planar graphs and maps 303 Contents xiii

15 Open problems 313 DAN ARCHDEACON 1. Introduction 313 2. Drawings and crossings 314 3. Genus and obstructions 317 4. Cycles and factors 320 5. Colourings and ﬂows 322 6. Local planarity 324 7. Thickness, book embeddings and covering graphs 325 8. Geometrical topics 328 9. Algorithms 330 10. Inﬁnite graphs 332

Notes on contributors 337 Index 341

Foreword

The origins of topological graph theory lie in the 19th century, largely with the four colour problem and its extension to higher-order surfaces – the Heawood map problem. With the explosive growth of topology in the early 20th century, mathematicians like Veblen, Rado and Papakyriakopoulos provided foundational results for understanding surfaces combinatorially and algebraically. Kuratowski, MacLane and Whitney in the 1930s approached the four colour problem as a question about the structure of graphs that can be drawn without edge-crossings in the plane. Kuratowski’s theorem characterizing planarity by two obstructions is the most famous, and its generalization to the higher-order surfaces became an influential unsolved problem. The second half of the 20th century saw the solutions of all three problems: the Heawood map problem by Ringel, Youngs et al. by 1968, the four colour problem by Appel and Haken in 1976, and finally the generalized Kuratowski problem by Robertson and Seymour in the mid-1990s. Each is a landmark of 20th-century mathematics. The Ringel–Youngs work led to an alliance between combinatorics and the algebraic topology of branched coverings. TheAppel–Haken work was the first time that a mathematical theorem relied on exhaustive computer calculations. And the Robertson–Seymour work led to their solution of Wagner’s conjecture, which provides a breathtaking structure for the collection of all finite graphs, a collection that would seem to have no structure at all. Each of these problems centres on the question of which graphs can be embedded in which surfaces, with two complementary perspectives – fixing the graph or fixing the surface.Although the question sounds highly focused, the study of graphs on surfaces turns out to be incredibly broad, rich in connections with other branches of mathematics and computer science: algorithms, computer-drawing, group theory, Riemann surfaces, enumerative combinatorics, block designs, finite geometries, Euclidean and non-Euclidean geometry, knot theory, the absolute Galois group, C*-algebras, and even string theory.

xv xvi Foreword

This volume attempts to survey the principal results within over-arching themes for the myriad aspects of topological graph theory. The authors of the chapters are recognized authorities in their fields. This book is written for the non-specialist and can be used as the basis for a graduate-level course. Nonetheless, the individual chapters cover their fields in great depth and detail, so that even specialists will find the book valuable, both as a reference and as a source of new insights and problems.

JONATHAN L. GROSS THOMAS W. TUCKER Preface

The field of graph theory has undergone tremendous growth during the past century. As recently as fifty years ago, the graph theory community had few members and most were in Europe and North America; today there are hundreds of graph theorists and they span the globe. By the mid-1970s, the field had reached the point where we perceived the need for a collection of surveys of the areas of graph theory: the result was our three-volume series Selected Topics in Graph Theory, comprising articles written by distinguished experts in a common style. During the past quarter-century, the transformation of the subject has continued, with individual areas (such as topological graph theory) expanding to the point of having important sub-branches themselves. This inspired us to conceive of a new series of books, each a collection of articles within a particular area written by experts within that area. The first of these books was our companion volume on algebraic graph theory, published in 2004. This is the second of these books. One innovative feature of these volumes is the engagement of academic consultants (here, Jonathan Gross and Thomas Tucker) to advise us on topics to be included and authors to be invited. We believe that this has been successful, the result being chapters covering the full range of areas within topological graph theory written by authors from around the world. Another important feature is that we have imposed uniform terminology and notation throughout, as far as possible, in the belief that this will aid readers in going from one chapter to another. For a similar reason we have not attempted to remove a small amount of overlap between the various chapters. We hope that these features will make the book easier to use in an advanced course or seminar. We heartily thank the authors for cooperating on this, even though it sometimes required their abandoning some of their favourite conventions – for example, many mathematicians use χ to denote the Euler characteristic, whereas for graph theorists χ usually denotes the chromatic number: the graph theorists won on this one. We also asked our contributors to undergo

xvii xviii Preface the ordeal of having their early versions subjected to detailed critical reading. We believe that the final product is thereby significantly better than it might otherwise have been, simply a collection of individually authored chapters. We want to express our sincere appreciation to all of our contributors for their cooperation. We extend special thanks to Jonathan Gross and Thomas Tucker for their willingness to share their expertise as academic consultants – their advice has been invaluable. We are also grateful to Cambridge University Press for publishing this work; in particular, we thank Roger Astley and Clare Dennison for their advice, support and cooperation. Finally, we extend our appreciation to several universities for the different ways in which they have assisted with this endeavour: the first editor is grateful to Indiana University–Purdue University in Fort Wayne, while the second editor has had the cooperation of the Open University and Keble College, Oxford.

LOWELL W. BEINEKE ROBIN J. WILSON Introduction

LOWELL W. BEINEKE and ROBIN J. WILSON

1. Graph theory 2. Graphs in the plane 3. Surfaces 4. Graphs on surfaces References

1. Graph theory

This section presents the basic deﬁnitions, terminology and notation of graph theory, along with some fundamental results. Further information can be found in the many standard books on the subject – for example, Chartrand and Lesniak [1], Gross and Yellen [2], West [3] or (for a simpler treatment) Wilson [4].

Graphs A graph G is a pair of sets (V, E), where V is a ﬁnite non-empty set of elements called vertices, and E is a ﬁnite set of elements called edges, each of which has two associated vertices (which may be the same). The sets V and E are the vertex-set and edge-set of G, and are sometimes denoted by V (G) and E(G). The order of G is the number of vertices, usually denoted by n, and the number of edges is denoted by m. An edge whose vertices coincide is called a loop, and if two vertices are joined by more than one edge, these are called multiple edges. A graph with no loops or multiple edges is a simple graph. In many areas of graph theory there is little need for graphs that are not simple, in which case an edge e can be considered as a pair of vertices, e ={v, w},orvw for simplicity. However, in topological graph theory, it is often useful, and sometimes necessary, to allow loops and multiple

1 2 Lowell W. Beineke and Robin J. Wilson

Fig. 1. edges. A graph of order 4 and its underlying simple graph are shown in Fig. 1. The complement G of a simple graph G has the same vertices as G, but two vertices are adjacent in G if and only if they are not adjacent in G.

Adjacency and degrees The vertices of an edge are incident with the edge, and the edge is said to join these vertices. Two vertices that are joined by an edge are neighbours and are said to be adjacent. The set N(v) of neighbours of a vertex v is its neighbourhood.Two edges are adjacent if they have a vertex in common. The degree deg v of a vertex v is the number of times that it occurs as an endpoint of an edge (with a loop counted twice); in a simple graph, the degree of a vertex is just the number of its neighbours. A vertex of degree 0 is an isolated vertex and one of degree 1 is an end vertex. A graph is regular if all of its vertices have the same degree, and is k-regular if that degree is k; a 3-regular graph is sometimes called cubic. The maximum degree in a graph G may be denoted by or (G), and the minimum degree by δ or δ(G).

Isomorphisms and automorphisms An isomorphism between two graphs G and H consists of a pair of bijections, one between their vertex-sets and the other between their edge-sets, that preserve incidence and non-incidence. (For simple graphs, this amounts to having a bijection between their vertex-sets that preserves adjacency and non-adjacency.) The graphs G and H are isomorphic, denoted by G ≈ H or G =∼ H , if there exists an isomorphism between them. An automorphism of a graph G is an isomorphism of G with itself. The set of all automorphisms of a graph G forms a group, called the automorphism group of G and denoted by Aut(G). A graph G is vertex-transitive if, for any vertices v and w, there is an automorphism taking v to w.Itisedge-transitive if, for any edges e and f , there is an automorphism taking the vertices of e to those of f . It is arc-transitive if, given two ordered pairs of adjacent vertices (v, w) and Introduction 3

(v, w), there is an automorphism taking v to v and w to w. This is stronger than edge-transitivity, since it implies that for each edge there is an automorphism that interchanges its vertices.

Walks, paths and cycles

A walk in a graph is a sequence of vertices and edges v0, e1, v1, ... , ek, vk,in which each edge ei joins the vertices vi−1 and vi. This walk goes from v0 to vk or connects v0 and vk, and is called a v0-vk walk. For simple graphs, it is frequently shortened to v0v1···vk, since the edges can be inferred from this. Its length is k, the number of occurrences of edges, and if v0 = vk, the walk is closed. Some important types of walk are the following: • a path is a walk in which no vertex is repeated; • a cycle is a non-trivial closed walk in which no vertex is repeated, except the ﬁrst and last; • a trail is a walk in which no edge is repeated; • a circuit is a non-trivial closed trail.

Connectedness and distance A graph is connected if there is a path connecting each pair of vertices, and disconnected otherwise. A (connected) component of a graph is a maximal connected subgraph. In a connected graph, the distance d(v,w)from v to w is the length of a shortest v-w path. It is easy to see that distance satisﬁes the properties of a metric. The diameter of a connected graph G is the maximum distance between two vertices of G.IfG has a cycle, the girth of G is the length of a shortest cycle. A connected graph is Eulerian if it has a closed trail containing all the edges of G; such a trail is an Eulerian trail. The following are equivalent for a connected graph G: • G is Eulerian; • the degree of each vertex of G is even; • the edge-set of G can be partitioned into cycles. A graph is Hamiltonian if it has a spanning cycle, and is traceable if it has a spanning path. No ‘good’ characterizations of these properties are known.

Bipartite graphs and trees If the set of vertices of a graph G can be partitioned into two non-empty subsets so that no edge joins two vertices in the same subset, then G is bipartite. The two 4 Lowell W. Beineke and Robin J. Wilson subsets are called partite sets and, if they have orders r and s, G is sometimes called an r × s bipartite graph. (For convenience, the graph with one vertex and no edges is also called bipartite.) Bipartite graphs are characterized by having no cycles of odd length. Among the bipartite graphs are trees, those connected graphs with no cycles. Trees have been characterized in many ways, some of which we give here. For a graph of order n, the following statements are equivalent:

• G is connected and has no cycles; • G is connected and has n − 1 edges; • G has no cycles and has n − 1 edges; • G has exactly one path between any two vertices.

A graph without cycles is called a forest; thus, each component of a forest is a tree. The set of trees can also be defined inductively: a single vertex is a tree; and for n ≥ 1, the trees with n + 1 vertices are those graphs obtainable from some tree with n vertices by adding a new vertex adjacent to precisely one of its vertices. This definition has a natural extension to higher dimensions. The k-dimensional trees,ork-trees for short, are defined as follows. The complete graph on k vertices is a k-tree, and for n ≥ k, the k-trees with n+1 vertices are those graphs obtainable from some k-tree with n vertices by adding a new vertex adjacent to k mutually adjacent vertices in the k-tree. Fig. 2 shows a tree and a 2-tree. An important concept in the study of graph minors (introduced later) is the tree-width of a graph G, the minimum dimension of any k-tree that contains G as a subgraph.

Fig. 2.

Special graphs We now introduce some individual types of simple graph:

• the complete graph Kn has n vertices, each adjacent to all the others; • the path graph Pn consists of the vertices and edges of a path of length n − 1; • the cycle graph Cn consists of the vertices and edges of a cycle of length n; Introduction 5

• the complete bipartite graph Kr,s is the simple r × s bipartite graph in which each vertex is adjacent to every vertex in the other partite set;

• in the complete k-partite graph Kr ,r ,...,r the vertices are in k sets with orders 1 2 k r1, r2, ..., rk, and each vertex is adjacent to every vertex in another set; if the k sets all have order r, the graph is denoted by Kk(r).

Examples of these graphs are given in Fig. 3.

P5: C5: K5:

K3(2): K3,3:

Fig. 3.

We also introduce some special graphs that are not simple:

• the bouquet Bm has one vertex and m incident loops; • the dipole Dm consists of two vertices with m edges joining them; • the cobblestone path is the 4-regular graph obtained from the path Pn by doubling each edge and adding a loop at each end.

Fig. 4 gives examples of these graphs.

B J 4: D4: 3:

Fig. 4.

A necklace is any graph obtained from a cycle by doubling each edge in an independent subset of its edges and adding a loop at each vertex that is not on 6 Lowell W. Beineke and Robin J. Wilson

Fig. 5. one of those edges. It is of type (r, s) if it has r doubled edges and s loops (so the original cycle has length 2r + s). The necklace in Fig. 5 is of type (3, 4).

Operations

Let G and H be graphs with disjoint vertex-sets V (G) ={v1,v2,...,vn} and V(H)={w1,w2,...,wr }: • the union G ∪ H has vertex-set V (G) ∪ V(H)and edge-set E(G) ∪ E(H). The union of k graphs isomorphic to G is denoted by kG; • the join G + H is obtained from G ∪ H by adding an edge from each vertex in G to each vertex in H; • the Cartesian product G H (or G × H) has vertex-set V (G) × V(H), and (vi, wj ) is adjacent to (vh, wk) if either vi is adjacent to vh in G and wj = wk, or vi = vh and wj is adjacent to wk in H: in less formal terms, G H can be obtained by taking n copies of H and joining corresponding vertices in different copies whenever there is an edge in G; • the lexicographic product (or composition) G[H] also has vertex-set V (G) × V(H), but with (vi,wj ) adjacent to (vh,wk) if either vi is adjacent to vh in G or vi = vh and wj is adjacent to wk in H.

Examples of these binary operations are given in Fig. 6.

Subgraphs and minors If G and H are graphs with V(H) ⊆ V (G) and E(H) ⊆ E(G), then H is a subgraph of G; if, moreover, V(H)= V (G), then H is a spanning subgraph. The Introduction 7

G ∪ H: G + H:

G × H: G[H]:

Fig. 6. subgraph S induced by a non-empty set of S of vertices of G is the subgraph H whose vertex-set is S and whose edge-set consists of those edges of G that join two vertices in S. A subgraph H of G is an induced subgraph if H = V(H).In Fig. 7, H1 is a spanning subgraph of G, and H2 is an induced subgraph.

G: H1: H2:

Graph Spanning subgraph Induced subgraph

Fig. 7.

Given a graph G, the deletion of a vertex v results in the subgraph obtained by removing v and all of its incident edges; it is denoted by G − v and is the subgraph induced by V −{v}. More generally, if S is any set of vertices in G, then G − S is the graph obtained from G by deleting all the vertices in S and their incident 8 Lowell W. Beineke and Robin J. Wilson edges – that is, G − S = V − S. Similarly, the deletion of an edge e results in the subgraph G − e and, for any set X of edges, G − X is the graph obtained from G by deleting all the edges in X. There are two other operations that are especially important for topological graph theory. If an edge e joins vertices v and w, the subdivision of e replaces e by a new vertex u and two new edges vu and uw. Two graphs are homeomorphic if there is some graph from which each can be obtained by a sequence of subdivisions. The contraction of e replaces the vertices v and w of e by a new vertex u, with an edge ux for each edge vx or wx in G. The operations of subdivision and contraction are illustrated in Fig. 8.

w Subdivision Contraction

v u

Fig. 8.

If a graph H can be obtained from G by sequence of edge-contractions and the removal of isolated vertices, then G is contractible to H .Aminor of G is any graph that can be obtained from G by a sequence of edge-deletions and edge-contractions, along with deletions of isolated vertices.

Connectedness and connectivity A vertex v of G is a cut-vertex if G − v has more components than G. A non- trivial connected graph with no cut-vertices is 2-connected or non-separable. The following statements are equivalent for a graph G with at least three vertices: • G is non-separable; • every pair of vertices lie on a cycle; • for any three vertices u, v and w, there is a u-w path containing v; • for any three vertices u, v and w, there is a u-w path not containing v. Introduction 9

More generally, G is k-connected if there is no set S of fewer than k vertices for which G−S is a connected non-trivial graph. Menger gave a useful characterization of such graphs:

Menger’s theorem (vertex version) A graph G is k-connected if and only if, for each pair of vertices v and w, there is a set of k internally disjoint v-w paths.

The connectivity κ(G) of a graph G is the maximum value of k for which G is k-connected. There are similar concepts and results for edges. A cut-edge (or bridge)isan edge whose deletion produces one more component than before. (Note: for some authors, ‘bridge’has a different meaning.) A non-trivial graph is k-edge-connected if the result of removing fewer than k edges is always connected, and the edge- connectivity λ(G) is the maximum value of k for which G is k-edge-connected. We note that Menger’s theorem also has an edge version:

Menger’s Theorem (edge version) A graph G is k-edge-connected if and only if, for each pair of vertices v and w, there is a set of k edge-disjoint v-w paths.

Graph colourings Agraph is k-colourable if, from a set of k colours, it is possible to assign a colour to each vertex in such a way that adjacent vertices always have different colours. The chromatic number χ(G) is the least value of k for which G is k-colourable, and if χ(G) = k, then G is k-chromatic. It is easy to see that a graph is 2-colourable if and only if it is bipartite, but there is no ‘good’ way to determine which graphs are k-colourable, for any k ≥ 3. Brooks’s theorem provides one of the best-known bounds on the chromatic number of a graph.

Brooks’s theorem If G is a simple graph with maximum degree and is neither an odd cycle nor a complete graph, then χ(G) ≤ .

There are similar concepts for colouring edges. A graph is k-edge-colourable if, from a set of k colours, it is possible to assign a colour to each edge in such a way that adjacent edges always have different colours. The chromatic index χ (G) is the least value of k for which G is k-edge-colourable. Vizing [11] proved that the range of values of χ(G) is quite limited:

Vizing’s theorem If G is a simple graph with maximum degree , then χ(G) = or + 1. 10 Lowell W. Beineke and Robin J. Wilson

Directed graphs Digraphs are directed analogues of graphs, and thus have many similarities, as well as some important differences. A digraph (or directed graph) D is a pair of sets (V, E), where V is a ﬁnite non- empty set of elements called vertices, and E is a set of ordered pairs of distinct elements of V called arcs or directed edges. Note that the elements of E are ordered, which gives each of them a direction. An example of a digraph is shown in Fig. 9.

v 2 v3

v1 v4

Fig. 9.

Because of the similarities between graphs and digraphs, we mention only the main differences here and do not redeﬁne those concepts that carry over easily. An arc (v, w) in a simple digraph may be written as vw, and is said to go from v to w,ortogo out of v and into w. In the context of digraphs, walks, paths, cycles, trails and circuits are understood to be directed, unless otherwise indicated. A digraph D is strongly connected if there is a path from each vertex to each of the others. A strong component is a maximal strongly connected subgraph. Connectivity and edge-connectivity are deﬁned in terms of strong connectedness.

2. Graphs in the plane

In this section we brieﬂy survey properties of graphs that can be drawn in the plane without any edges crossing. To make this more precise, we deﬁne an embedding of a graph G in the plane to be a one-to-one mapping of the vertices of G into the plane and a mapping of the edges of G to disjoint simple open arcs, so that the image of each edge joins the images of its two vertices and none of the images of the edges contains the image of a vertex. Here there is little to be gained by allowing loops or multiple edges, so in this section we assume that all graphs are simple. A graph that can be embedded in the plane is called a planar graph, and its image is called a plane graph. An example is given in Fig. 10. A region of an embedded graph G is a maximal connected set of points in the relative complement of G in the plane; note that one region is unbounded. The topological closure of a region (that is, the region together with the vertices and Introduction 11

r4 r1

Fig. 10. edges of G on its boundary) is a face. If a face has a connected boundary, that boundary is a closed walk, and the length of that walk is the size of the face. Steinitz showed that 3-connected planar graphs form a particularly nice class of graphs:

Steinitz’s theorem A graph is the skeleton of a polyhedron if and only if it is 3-connected and planar.

The fundamental theorem on planar graphs is an extension of Euler’s well- known formula for polyhedra:

Euler’s formula If an embedding of a connected graph in the plane has n vertices, m edges and r regions, then n − m + r = 2.

One consequence of Euler’s formula is that a planar graph G with n (≥ 3) vertices has at most 3(n − 2) edges, and at most 2(n − 2) if G is bipartite. It is easy to deduce from these observations that the graphs K5 and K3,3 are non-planar. Kuratowski proved that these are in fact the only barriers to planarity:

Kuratowski’s theorem A graph is planar if and only if it does not contain a subgraph homeomorphic to K5 or K3,3.

There are other criteria for planarity, but we mention only one here; it is due to Wagner and simply has the word ‘contractible’ in place of ‘homeomorphic’:

Wagner’s theorem A graph is planar if and only if it does not contain a subgraph contractible to K5 or K3,3.

There are several measures of non-planarity, two of which are as follows: • the crossing number of a graph G is (informally, but intuitively clear) the minimum number of crossings of pairs of edges in any drawing of G in the plane; • the thickness of G is the minimum number of graphs in a set of planar graphs whose union is G. 12 Lowell W. Beineke and Robin J. Wilson

3. Surfaces

Much of the interest in topological graph theory involves graphs on surfaces other than the plane. In this section we say what is meant by a surface; the topic of embeddability on these surfaces is introduced in the next section. A surface is a topological space with the following two properties:

• it is a 2-manifold – that is, each point has a neighbourhood homeomorphic to an open disc; • it is compact – that is, it is closed and bounded.

Note that this deﬁnition is quite restrictive – the plane does not qualify as a surface in this sense since it is not compact, and the Möbius strip does not qualify since it has a boundary and thus is not compact. A surface is orientable if a positive sense of rotation (say, clockwise) can be made around all points consistently, and is non-orientable otherwise. The simplest orientable surfaces are the sphere and the torus (Fig. 11), while the simplest non- orientable surfaces are the projective plane and the Klein bottle (Fig. 12).

Fig. 11.

Fig. 12.

There are two ways to increase the complexity of a surface: by adding either a handle or a crosscap. To add a handle to a surface S, remove two disjoint open discs from S and identify their boundaries with the ends of a truncated cylinder in a consistent manner (see Fig. 13). To add a crosscap to S, remove one open disc and identify its boundary with that of a Möbius strip (see Fig. 14); this is equivalent to identifying opposite points on the boundary of the disc. It is a fact that no matter how h handles are added to the sphere, the result is effectively the same: an orientable surface that we denote by Sh. Similarly, no matter how k crosscaps (k > 0) are added to the sphere, the result is effectively the Introduction 13

Fig. 13.

Fig. 14. same: a non-orientable surface that we denote by Nk. Furthermore, every surface is homeomorphic either to Sh for some h ≥ 0, or to Nk for some k ≥ 1. When the number of handles or crosscaps on a surface is small, it can be useful to represent it as a polygon. For an orientable surface Sh, take a 4h-gon and identify its sides according to the pattern

α β α−1 β−1 ... α β α−1 β−1 1, 1, 1 , 1 , , h, h, h , h .

Fig. 15 shows the torus and the double torus represented in this way.

b a 2 1

b a 1 a 2

b b a b 1 2

a b a 2 1

Torus Double-torus

Fig. 15. 14 Lowell W. Beineke and Robin J. Wilson

For the non-orientable surface Nk, take a 2k-gon and identify its sides according to the pattern α1, α1, α2, α2, ... , αk, αk. Fig. 16 shows the projective plane and the Klein bottle represented in this way.

a a 2 1

Projective plane Klein bottle

Fig. 16.

4. Graphs on surfaces

Many of the concepts we introduced in Section 2 carry over in a natural way to graphs on surfaces in general. Speciﬁcally, a graph on a surface S is the analogue of a plane graph, and a graph G is embeddable on S if it is isomorphic to a graph on that surface; we refer to this as an embedding of G. An embedding is cellular if every region is homeomorphic to an open disc. Figs. 17 and 18 show cellular embeddings of the complete graphs K6 and K7 on the projective plane and the torus. Regions and faces are deﬁned as for the plane. A cellular embedding in which each face has three sides is a triangulation, and one in which each face has four sides is a quadrangulation. (In this section all graphs are assumed to be simple.) The embeddings in Figs. 17 and 18 are both triangulations. Every surface has a version of Euler’s polyhedron formula:

Euler’s formula If a simple graph G has a cellular embedding in a surface S with n vertices, m edges and r regions, then − h S = S , n − m + r = 2 2 if h 2 − k if S = Nk.

The number associated with S in this theorem is called its Euler characteristic, denoted by ε(S). Introduction 15

b c

a d

e e

d a

c b

Fig. 17.

abcda

e e

f f

g g

abcda

Fig. 18.

The genus of a graph The genus γ (G) of a graph G is the minimum genus of a surface in which the graph can be embedded – that is, the minimum number of handles that need to be added to the sphere for G to be embeddable. It follows from Euler’s formula γ (G) 1 m − 1 n + that has the general lower bound of 6 2 1; this can be improved to 1 m − 1 n + G 4 2 1if is bipartite. In part because of the connection with colouring maps on surfaces, much of the focus of early work was on the genus of complete graphs. The solution to this difﬁcult problem was ﬁnally completed by Ringel and Youngs (see [7]) in 1968:

Ringel–Youngs theorem For n ≥ 3, the genus of the complete graph Kn is

γ(K ) = 1 (n − )(n − ) . n 12 3 4

An important consequence of this result is that it yields the chromatic number of every orientable surface other than the plane. 16 Lowell W. Beineke and Robin J. Wilson

The crosscap number of a graph The non-orientable analogue of the genus of a graph is the non-orientable genus (or crosscap number) γ (G), the minimum number of crosscaps that need to be added to the sphere for G to be embeddable. In 1954, Ringel determined the non-orientable genus of the complete graph: Ringel’s theorem For n ≥ 3, the non-orientable genus of the complete graph Kn is γ(K ) = 1 (n − )(n − ) , n 6 3 4 except that γ(K7) = 3.

There are some interesting comparisons to be made between the parameters γ (G) and γ (G). Since any surface with a crosscap is non-orientable, it follows that, for any graph G, γ (G) ≤ 2γ (G) + 1. There is no bound in the other direction, however, as there are graphs of arbitrarily large orientable genus that can be embedded in the projective plane. Other differences appear in cellular embeddings of graphs. If γ (G) = h, then every embedding of G on Sh is cellular, but the corresponding statement for the non-orientable genus does not hold. In particular, although γ(K7) = 3, not every embedding of K7 in N3 is cellular. Furthermore, while the orientable genus is additive over the blocks of a graph the non-orientable genus is not – the graph consisting of two copies of K7 is a counter-example.

The chromatic number of a surface Much of the interest in embedding complete graphs is related to colourings of maps and graphs. The chromatic number χ(S)of a surface S is the maximum chromatic number among all S-embeddable graphs. As Heawood noted, a lower bound for χ(S)can be deduced from Euler’s formula. For all surfaces other than the sphere, the sharpness of this bound follows from the genus and the non-orientable genus of complete graphs (and a little more for the Klein bottle).

Map colour theorem Except for the Klein bottle N2, which has chromatic number 6, the chromatic number of a surface S of Euler characteristic ε is √ χ(S) = 1 + − ε . 2 7 49 24

Kuratowski-type theorems

Every surface has a family of graphs that plays the role of K5 and K3,3 in Kuratowski’s theorem. A minor-minimal forbidden family M(S) of a surface S is a set of graphs with these three properties: Introduction 17

• no graph in M(S) is embeddable in S; • every graph that is not embeddable in S has a graph in M(S) as a minor; • no graph in M(S) is a minor of another graph in M(S). There is a corresponding (and larger) family if, instead of minors, homeomorphic subgraphs are considered. The projective plane is the only surface other than the sphere for which these families are known: the minor-minimal family contains 35 graphs and the homeomorphically irreducible family contains 103. One of the foremost results in topological graph theory is that these families are always ﬁnite. For non-orientable surfaces this was established by Archdeacon and Huneke in 1980. The orientable case was not settled until 1984, when Robertson and Seymour proved their spectacular result on graph minors.

Robertson–Seymour theorem Every inﬁnite collection of graphs contains at least one graph that is a minor of another.

Corollary The set of minor-minimal forbidden graphs of every surface is ﬁnite.

References

1. G. Chartrand and L. Lesniak, Graphs and Digraphs (4th edn), Chapman & Hall/CRC, 2004. 2. J. L. Gross and J. Yellen, Graph Theory and its Applications (2nd edn), Chapman & Hall/CRC, 2005. 3. D. B. West, Introduction to Graph Theory (3rd edn), Prentice-Hall, 2007. 4. R. J. Wilson, Introduction to Graph Theory (4th edn), Pearson, 1996. 1 Embedding graphs on surfaces

JONATHAN L. GROSS and THOMAS W. TUCKER

1. Introduction 2. Graphs and surfaces 3. Embeddings 4. Rotation systems 5. Covering spaces and voltage graphs 6. Enumeration 7. Algorithms 8. Graph minors References

In this ﬁrst chapter, we review the basic ideas of topological graph theory. We describe the principal early theme of constructing embeddings, and we then survey the launching of the dominant programmatic themes of the present era, which are presented in greater detail individually in subsequent chapters.

1. Introduction

By the late 19th century, the work of Heawood [16] and Heffter [17] had expanded the study of graph drawings beyond the conﬁnes of the plane to surfaces of higher order. Over the next hundred years or so, the solution of several long-standing problems attracted many researchers and the present-day programmatic themes were set into place. Of course, some of the methods used in the solutions led to new problems. Topological graph theory is now one of the largest branches of graph theory. This chapter gives a brief overview of some of the principal concepts, terminology and notation of topological graph theory. As general resources, we recommend [13], Chapter 7 of [14], [22] and [44].

18 1 Embedding graphs on surfaces 19

2. Graphs and surfaces

We start by recalling some definitions from the Introduction.Agraph G is formally defined to be a combinatorial incidence structure with a vertex-set V and an edge- set E, where each edge e is incident with at most two vertices; we may write VG and EG, respectively, when there is more than one graph under consideration. A graph may have multiple adjacencies and loops and is usually taken to be finite unless the immediate context implies otherwise. In some contexts, the letters n and m are reserved for the number |V | of vertices and the number |E| of edges. The underlying topological space of a graph, also commonly called ‘the graph’, is the 1-dimensional cell-complex with a 0-cell for each vertex v and a 1-cell for each edge e. In that sense, each edge has two edge-ends and two endpoints (which may coincide if the edge is a loop). In some contexts, edges are assigned a direction from one endpoint to the other, usually indicated in a drawing by an arrow on an edge. The degree, also called the valence, of a vertex v is the number of edge-ends incident to v, that is the number of incident edges with loops weighted twice. A graph is called regular if all vertices have the same degree. A surface S is a connected topological space such that every point has an open neighborhood homeomorphic to the interior of the unit disc (this definition does not allow a surface to have boundary). A surface is closed if is compact, and is orientable if it contains no subset homeomorphic to the Möbius band (the space obtained by identifying a pair of opposite sides of a rectangle as shown in Fig. 1). Just as the plane has two orientations, clockwise and anticlockwise, an orientable surface has two possible orientations; if one is specified, we say that the surface is oriented.

Fig. 1. Constructing a Möbius band

Every surface has a triangulation [27] into homeomorphic copies of a triangle, which intersect only along entire edges or at vertices. This so-called piecewise- linear structure of a surface as a 2-complex is unique up to piecewise-linear homeomorphism [24], which means that surfaces can be treated as combinatorial objects, as well as topological ones. The Introduction to this volume describes both the orientable surfaces Sh and the non-orientable surfaces Nk. 20 Jonathan L. Gross and Thomas W. Tucker

3. Embeddings

An embedding of a graph G on a surface S is a continuous function f : G → S taking G homeomorphically to its image f (G). Intuitively, an embedding is a drawing of a graph on a surface in which no edges cross. The components of the complement of the image of an embedded graph are called regions. An embedding is cellular if every region is homeomorphic to an open disc. In nearly every aspect of topological graph theory, the only embeddings considered are cellular. Consequently, the word ‘cellular’ is usually omitted by most authors, as it will be in this book. Since an embedding of a non-connected graph cannot be cellular, it is often implicit from the context that an embedded graph is connected. For a cellular embedding, the topological closure of a region is called a face. The set of faces is denoted by F , with subscripts as needed if more than one embedding is under consideration. The closed walk in the underlying graph G corresponding to the boundary of a region is called a boundary walk or face boundary walk of the corresponding face, and is unique up to the choice of the initial vertex and the choice of orientation of the region. The length of this boundary walk is called the size of the face. An embedding is strongly cellular or circular if every face is homeomorphic to a closed disc – that is, if every boundary walk is a cycle. If G is not 2-connected, then it has no strongly cellular embedding. The converse result, that every 2-connected graph has a strongly cellular embedding, is known as the circular embedding conjecture and is open at this time. It implies the closely related cycle double cover conjecture (of [33] and [34]) that every 2-edge-connected graph has a collection of cycles that includes every edge exactly twice. Chapter 15 provides further information about these conjectures as part of a large collection of open problems. The orientable genus range of a graph G is the set of integers h (which are easily proved to be consecutive) such that the graph G is cellularly embeddable in the surface Sh. The minimum of this range is called the minimum genus of the graph (or often simply, the genus). The maximum is called the maximum genus. The minimum and maximum genus of the graph G are denoted by γmin(G) (or often, simply γ (G)) and γmax(G), respectively. A graph of genus 0 is said to be planar. The crosscap range of a graph G is the set of integers k (also easily proved to be consecutive) such that the graph G is cellularly embeddable in the surface Nk. The minimum of this range is called the minimum crosscap number of the graph (or, often, simply the crosscap number). The maximum is called the maximum crosscap number. The minimum and maximum crosscap numbers of the graph G are denoted by γ˜min(G) and γ˜max(G), respectively. 1 Embedding graphs on surfaces 21

The Poincaré dual embedding for a cellular graph embedding G → S (called the primal embedding in this context) is constructed as follows: • in the interior of each primal region, a dual vertex is drawn; • through each primal edge, a dual edge is drawn joining the dual vertex on one side of the edge to the dual vertex on the other (thus, a loop whenever the same primal region lies on both sides of that primal edge); • if the surface S is oriented, then in the dual embedding, the orientation is reversed.

Aﬂat-polygon representation of an embedding K4 → S1 and its dual embedding are shown in Fig. 2. The primal vertices and the primal edges are solid, the dual vertices are hollow, and the dual edges are dashed. Observe that four edges join the two dual vertices and that one of the dual vertices has two loops incident with it. There are many other ways to draw a graph embedding or even a graph (in 3-space, with edge-crossings in the plane, with straight lines as edges, with ovals as edges, etc). Chapter 8 surveys the different ways that one can try to visualize a graph or embedding.

b b

Fig. 2. A toroidal embedding and its dual

The Euler polyhedron formula for a cellular embedding of a graph is 2 − 2h for the orientable surface Sh, |V |−|E|+|F |= 2 − k for the non-orientable surface Nk.

The value of the expression on the right side of the equation is called the Euler characteristic of the surface, denoted by ε. The special case |V |−|E|+|F |=2 for the sphere, first stated by Euler in 1750, is reasonably regarded as the first result of topological graph theory, even though various topological aspects, such as the Jordan curve theorem, the Schoenfliess theorem, and the triangulability of surfaces, were not proved until early in the 20th century. (There are many proofs; see, for example, [13].) 22 Jonathan L. Gross and Thomas W. Tucker

Amap on a surface S is another name for an embedding G → S of a graph into S and is used when the focus is on the symmetries of the underlying vertex–edge–face incidence structure. Whereas embedding theory tends to rely on methods inspired by topological intuition, map theory depends more on group theory, especially on permutation groups. Viewing a map as an incidence structure, especially when all faces are triangles, leads naturally to designs and triple systems, as surveyed in Chapter 13. It also leads to the point–line incidence structure of ﬁnite geometries, which is covered in Chapter 12. An isomorphism of graphs is a bijection of the vertex-sets and of the edge-sets that respects the incidence structure. A homeomorphism of graphs as topological spaces induces a graph isomorphism, but there are many homeomorphisms inducing the same graph isomorphism (just as there are many homeomorphisms of the unit interval taking 0 → 0 and 1 → 1). An isomorphism between a graph and itself is called an automorphism; the set of all automorphisms of a graph G forms a group, denoted by Aut(G). An isomorphism of graph embeddings G → S and G → S is an isomorphism of the underlying graphs that takes face boundaries to face boundaries. A homeomorphism from S to S taking the image of G to the image of G induces an isomorphism of the embeddings, but again there are many such homeomorphisms inducing the same isomorphism. An isomorphism between an embedding G → S and itself is called an automorphism, and the set of all such automorphisms is a subgroup of Aut(G).

Planarity and colouring The development of topological graph theory as a distinct branch of graph theory was motivated by two problems regarding planar graphs. The first problem is concerned with map-colouring. The chromatic number of a map is most conveniently defined to be the chromatic number of the dual graph for that map – that is, as the smallest number of colours needed to colour the dual vertices so that distinct endpoints of a dual edge get different colours. The chromatic number of a surface is the maximum of the set of numbers that occur as chromatic numbers of maps on that surface or, equivalently, as the maximum of the set of numbers that occur as chromatic numbers of graphs on that surface. The four-colour problem is to prove that every planar map has chromatic number at most 4. Its first known written mention is in a letter from De Morgan to Hamilton in 1852. It was solved by Appel and Haken [2] in 1976. The second problem, called the planarity problem, is to characterize the graphs that are planar. The solution by Kuratowski [20] characterizes them completely in terms of two forbidden types of subgraphs – that is, homeomorphic copies of the complete graph K5 and the complete bipartite graph K3,3. The generalization of 1 Embedding graphs on surfaces 23

Kuratowski’s theorem seeks, for each surface, a complete ﬁnite set of obstructions to embeddability in that surface. On the other hand, if a graph is not planar, rather than embed it in a higher genus surface, we may still want to draw it in the plane, yet now allowing edge-crossings. The crossing number of a graph G can be described intuitively as the minimum number of edge-crossings in a drawing of G in the plane; it is another measure of the extent to which a graph fails to be planar. After starting some decades ago with the determination of a few difﬁcult special cases, this pursuit has emerged into a more general topic, involving forbidden subgraphs, as described in Chapter 7.

4. Rotation systems

Besides finding a flaw in Kempe’s attempted proof of the four-colour theorem, Heawood [16] expanded the quest to finding the colouring numbers of all closed surfaces. In 1890, he proved that the quantity √ H(ε) = 1 + − ε 2 7 49 24

– which is now called the Heawood number of a surface of Euler characteristic ε – is an upper bound for the chromatic number of the surface. Proving that it cannot be improved for any surface except the Klein bottle N2 became known as the Heawood problem. Its solution, completed in 1968 by Ringel and Youngs [30], required the construction of minimum-genus embeddings for the complete graphs. The construction employs a combinatorial method for specifying those embeddings that originated with Heffter. In an embedding in an oriented surface, the rotation at a vertex is the cyclic ordering of the edge-ends incident to that vertex, induced by the speciﬁed orientation. The set of all rotations is called the rotation system (or rotation scheme). Detailed examples of rotation systems are given, for instance, in Section 3.2 of [13] and by Section 6.6 of [44]. It is not hard to see how to trace out the face boundaries of the embedding using only the rotation system. Thus, every rotation system on a graph – that is, an assignment of a cyclic order to the edge-ends incident to each vertex – determines an oriented embedding. In his proof that the Heawood bound is achievable for the orientable surfaces Sh with h = 1, 2,...,6, Heffter [17] listed the vertices on the boundary walk of each face. This form of embedding speciﬁcation was dualized by Edmonds [5] into the form more widely used and led to the ‘Edmonds algorithm’ of determining the minimum genus of a graph by inspecting all its possible rotation systems. 24 Jonathan L. Gross and Thomas W. Tucker

A rotation system can be encoded by two permutations of the set of directed edges: a permutation ρ whose cycles are the rotation of the directed edges beginning at each vertex, and an involution λ that takes each directed edge to its reverse. To trace the face boundaries, one merely computes the cycles of ρλ. The permutation group generated by ρ and λ (with the specification of these two generators) is called the monodromy group or dart group of the map. Thus, we might view an oriented embedding as no more than a permutation group with two specified generators, one of which is an involution without fixed points. Although this viewpoint removes geometrical intuition, it is particularly helpful for computer construction, for instance, of all highly symmetric maps of low genus (see [4]). Viewing an embedding as a permutation group on the set of directed edges introduces terminology, notation and techniques from the theory of groups acting on sets. A (left) action of the group A on the set W is a homomorphism from A into the group of permutations on the set W, where we write permutations on the left aw (prefix notation). If the homomorphism is injective, the action is said to be faithful. For a faithful action, if aw = w for all w ∈ W, then a is the identity. The stabilizer of a ∈ W, denoted by Aw, is the subgroup of all a ∈ A such that aw = w. An action of A on W is transitive if, for all w, z ∈ W there is an a ∈ A such that aw = z. A transitive action is regular if the stabilizer Aw is trivial: it is easy to show that all stabilizers for a transitive action are conjugate in A,soifone is trivial, they are all trivial. Given an oriented embedding, an automorphism of the graph is an automorphism of the embedding if and only if the graph automorphism either preserves the rotation at every vertex, or reverses the rotation at every vertex. In the former case, the automorphism is orientation-preserving and in the latter case it is orientation-reversing. Rotation systems for non-orientable embeddings are more complicated. First we choose one of the two possible cyclic orderings of the edges incident to each vertex induced by the embedding. Then each edge is assigned one of two possible types, flat or twisted (alternatively called type-0 and type-1, or sign + and sign −), depending on whether or not an open neighbourhood of the edge can be given an orientation consistent with the rotations at its endpoints. It is not hard to use the information of vertex rotations and edge types to trace the face boundary walks, so any such assignment of a general rotation system to a graph defines an embedding of the graph. In this case, however, apparently different general rotation systems can define the same embedding: we can always choose to reverse the cyclic order at a vertex in exchange for reversing the type of all edges incident to the vertex. Two rotation systems are equivalent if we can get from one to the other by a sequence of such moves. Notice that a general rotation system for an orientable embedding may have twisted edges.Ageneral rotation system defines an orientable embedding if and only if each cycle contains an even number of twisted edges or, 1 Embedding graphs on surfaces 25 equivalently, if and only if there is an equivalent general rotation system for which all edges are flat. Automorphisms of embeddings can be interpreted in terms of general rotation systems as follows. Given a general rotation system for a graph G, a graph automorphism f of G gives an automorphism of the associated embedding if and only if there is an equivalent general rotation system such that f either preserves all vertex rotations or reverses all vertex rotations and f preserves edge types. Coding general embeddings as permutation groups uses ideas of Tutte [41]. Given a map, add a vertex at the midpoint of each edge and at the centre of each face, and then subdivide the map into triangles by adding edges from each original vertex to each new vertex in an incident face and from each new edge-vertex to each of the two new vertices in the faces incident to that edge, as indicated in Fig. 3. (The process constructs the first barycentric subdivision of a 2-complex.)

Fig. 3. Using ﬂags to encode an embedding

The resulting triangles are called flags and the embedding is determined when we specify three involutions on the set of flags, telling us which pairs of sides of flags to identify: T for vertex-edge sides, L for edge-face sides, and R for vertex-face sides. Since each edge lies on a diamond of four flags identified alternately along vertex-edge sides and edge-face sides, the permutation LT LT is the identity. Thus, we can view a map as a permutation group, called again the monodromy group, generated by three fixed-point-free involutions T,L,R satisfying LT LT = 1. A map given by such a permutation group is orientable if and only if the subgoup of the monodromy group consisting of even length words in T,L,R has index 2 (it necessarily has index at most 2). This abstract group- theoretic view of maps proves useful in computer constructions. The idea of a map as a group is developed in Chapter 11, with particular attention to maps supporting regular actions by automorphisms on directed edges, flags or vertices. 26 Jonathan L. Gross and Thomas W. Tucker

5. Covering spaces and voltage graphs

In Ringel’s initial work on the Heawood problem in the early 1950s, he used rotation systems and surgery – that is, modifying an embedding by cutting apart the embedding surface and sewing on a handle or crosscap to allow the addition of extra edges. Surgery was a well-established technique by the early 20th century (see [32]). Ringel’s innovation was to amplify the power of rotation systems by using one, two or three rotations algebraically to generate the remainder of the system. He masterfully designed generating rotations that would solve intricate special problems. After Gustin [15] created a remarkable computational aid (now called a combinatorial current graph) to construct generating rotations, Youngs joined Ringel in a five-year pursuit of the complete solution, which involved many different forms of combinatorial current graph, each defined by different rules. The complete solution, announced in [30], occupied about 300 journal pages. Topologically, a covering (or covering projection) p: X˜ → X of surfaces or graphs is a continuous surjection satisfying this condition: each point x of the codomain X has a neighbourhood U such that every component of p−1(U) is mapped homeomorphically onto U. For a surface, if this condition holds everywhere except for a finite number of points of the codomain, then the mapping is called a branched covering; its topological abstraction from Riemann surfaces can be traced back to Alexander [1] and A. W. Tucker [37]. Of particular interest are the regular coverings, which are obtained from the action of a group A of automorphisms on X˜ , where X = X/˜ A, the quotient space of orbits of A, and where p: X˜ → X/A is the natural quotient projection. For graphs we need the action to be fixed-point-free to get a covering, and for surfaces we need the action to be fixed-point-free except at a finite number of points to get a branched covering. For other group actions, the orbit space has the structure of an ‘orbifold’, a concept that plays a key role in Thurston’s study of geometrical structures on 3-manifolds (see [36]). Whereas the topological theory of covering spaces describes an existential relationship between the domain and the codomain of a mapping, the theory of voltage graphs, due to Gross [7] and Gross and Tucker [12], provides a combinatorial tool for constructing graphs and graph embeddings. In voltage graph theory, the many specialized forms of combinatorial current graph originating with Gustin and augmented by Ringel and Youngs (see [29]) are all unified, so that the Ringel–Youngs embeddings are readily understood as the duals of coverings of voltage graphs (see [9] and [11]). Moreover, the power of the technique was amplified so that it applies not only to embeddings of complete graphs, but to any embedding with sufficient symmetry. 1 Embedding graphs on surfaces 27

One view of a covering is that every cycle in the base graph unwinds when lifted to the covering graph. It is natural, therefore, to look at coverings where cycles unwind completely, just as the unit circle unwinds to the real line when lifted via the covering eit. Thus, every finite graph that is not a tree is covered by an infinite tree, where vertices repeated by a walk in the graph may now become separate when the walk is lifted to the tree. Coverings lead inevitably to infinite graphs, which have a far more complicated topological structure, whether embedded or not. Chapter 14 gives an introduction to the topology of infinite graphs.

Regular voltages Given a digraph G = (V, E),aregular voltage assignment in a group B is a function α: E → B that labels each edge e with a value α(e): • the pair G, α is called a regular voltage graph; • the graph G is called the base graph; • the group B is called the voltage group; • the label α(e) is called the voltage on the edge e. It is said to specify the covering digraph Gα, deﬁned as follows: • V(Gα) = V α = V × B, the Cartesian product; • E(Gα) = Eα = E × B; • if the edge e is directed from a vertex u to a vertex v in G, then the edge α eb = (e, b) in G is directed from the vertex ub = (u, b) to the vertex vbα(e) = (v, bα(e)). A more general concept, called a permutation voltage graph, was introduced in [12]. Whereas every regular covering, in the sense of topology, is realizable by a regular voltage assignment, all coverings (including the non-regular coverings) are realizable by permutation voltage assignments. Vertices and edges of the covering graph are usually speciﬁed in a subscripted notation, rather than in Cartesian product notation. There is a standard exception to this convention, intended to avoid double subscripting. The digraph Gα is usually called simply the covering graph. Moreover, its underlying (undirected) graph is also denoted by Gα and is also called the covering graph. Using such shared terminology avoids excessively formal prose; in context, no ambiguity results. Fig. 4(a) shows a regular voltage graph G, α: E → Z3, and Fig. 4(b) shows the corresponding covering graph. Assigning an involution x as the voltage to a loop e at a vertex v in the base graph causes the e-edges in the covering digraph to be paired – that is, the directed edge eb from a vertex vb to a vertex vb+x is paired with the directed edge eb+x from vb+x to vb. The term covering graph also refers to the undirected graph that is 28 Jonathan L. Gross and Thomas W. Tucker

u2 b2 v2 c c1 a a1 2 0 b a2 1 u v u1 b1 v1 c 1

a0 voltages in Z3 c0

u0 b0 v0

(a) (b)

Fig. 4. A regular voltage assignment and the covering graph obtained by identifying these pairs of directed edges as a single edge, even though that usage conﬂicts with standard topological usage. (The present authors used ‘derived graph’ in their earlier papers to avoid this discrepancy in usage.) Let G = (V, E), α: E → B be a regular voltage graph. The graph mapping from the covering graph Gα to the voltage graph G, given by the vertex function and edge function vb → v and eb → e respectively, is called the natural projection. (Thus, the natural projection is given by ‘erasure of subscripts’.) Let Gα be the covering graph for a regular voltage graph G = (V, E), α: E → B. Then

• the vertex subset {v}×B = {vb: b ∈ B} is called the (vertex) ﬁbre over v; • similarly, the edge subset {e}×B = {eb: b ∈ B} is called the (edge) ﬁbre over e.

An assignment of voltages to a graph also induces an assignment of voltages to directed walks, simply by taking the product of the voltages. Many of the properties of the covering graph – for example, whether it is connected – can be stated in terms of the voltages assigned to closed walks, all starting from a root vertex. If the base graph is a bouquet and the set X of voltages appearing on edges generates the voltage group A, then the covering graph is a Cayley graph for the group A, denoted C(A,X). The usual definition of C(A,X) is the graph with vertex-set A and a directed edge labelled x from a to ax, for every a ∈ A and every x ∈ X; left multiplication by A provides the action of A on C(A,X) by automorphisms. The graph underlying C(A,X), without directions and labels and with parallel edges identified, is also called a Cayley graph and is denoted C(A,X). White [43] defined the genus of a group as the minimum genus taken over all its Cayley graphs. Proulx [25] classified the toroidal groups, which fall into 17 infinite families plus some sporadic cases, and Tucker [38] established that there are finitely many groups of each genus greater than 1 and only one of genus 2. 1 Embedding graphs on surfaces 29

The faces of an embedding of a Cayley graph C(A,X) in a surface provide relators: words in the generators that reduce to the identity. The hope of Burnside, Dehn and others, when they ﬁrst studied embeddings of Cayley graphs 100 years ago, was that these geometrical pictures of a group might reveal the algebraic structure of a group via a presentation in terms of generators and relators. Such a presentation is given in a format like this:

x,y,z: x2 = y2 = z2 = 1,(xy)2 = (yz)3 = (xz)4 = 1.

There are various logical difficulties with such presentations, the most famous one (established by Michael Rabin [26]) being the non-existence of an algorithm to decide whether any given presentation defines the trivial group. On the other hand, a group presentation is an extremely efficient way of describing a group and is the basis for all computer calculations for groups (and therefore maps).

Lifting embeddings When a voltage graph is cellularly embedded in a surface, we obtain an embedding for the covering graph by using the rotation system for the base embedding to define one for the covering graph: vertex rotations and edge types are the same as those in the base graph with subscripts erased. Moreover, the graph projection extends to a branched covering from the covering embedding to the base embedding, with branching inside any face whose boundary walk has non-identity voltage. (See [8] for a more complete description and illustrative examples.) When the base graph is a bouquet and the voltages X generate the voltage group A, the resulting embedding is called a Cayley map (see [3] and [28]), which can be denoted by CM(A,ρ), where ρ is given as a single cycle of the elements of X and their inverses. For example, CM(Z7,(1, 3, 2, −1, −3, −2)) defines a triangular embedding of K7 in the torus having a fixed-point free action of Z7. The minimum genus over all Cayley maps for a group, which is called the strong symmetric genus, is always at least as large as the genus (see Tucker [39]) and is easier to compute in many cases. A survey of the genus and other group parameters like the strong symmetric genus is given in Chapter 10.

6. Enumeration

Enumeration is a classical pursuit in mathematics, and the development of powerful counting methods for graphs preceded their adaption to counting topological objects. Calculating surface-by-surface inventories of embeddings of a given graph, programatically initiated by Gross and Furst [10], combines some of the principal methods of embedding construction, especially rotations and surgery, 30 Jonathan L. Gross and Thomas W. Tucker with a variety of standard enumerative methods. Such inventories are the topic of Chapter 3. In recent years, Kwak and Lee have led in the application of voltage graph methods for enumerating graph coverings, and Chapter 9 provides an account of this active branch of topological graph theory. Combinatorial methods predominated in the older, complementary programme of research launched by Tutte [40], [41] into the counting of maps on a given surface. Jackson and Visentin [19] have provided a complete listing of the maps with a small number of edges.

7. Algorithms

The prototypical embeddability problem in topological graph theory is to determine a formula for the minimum genus of the graphs in an inﬁnite class. The outstanding single example is the Ringel–Youngs formula γ (K ) = 1 (n − )(n − ) min n 12 3 4 for the genus of a complete graph. Ringel also derived formulas for the minimum genus of hypercube graphs and of complete bipartite graphs. With such success, there arose the question of the existence of a polynomial-time algorithm to calculate the minimum genus. Planarity testing may have seemed an initial step toward the more general goal. There were informal methods that rather quickly made the planarity decision, and some formal quadratic-time algorithms were developed. The naive algorithm based on this Kuratowski characterization does not run in polynomial time. Nonetheless, when iterative application of the Jordan curve theorem is included in the test, a quadratic-time algorithm is achievable. Ultimately, Hopcroft and Tarjan [18] produced a linear-time algorithm. Eventually, Thomassen [35] showed that deciding whether a given graph has a given minimum genus is an NP-complete problem. Interestingly, Mohar [21] subsequently showed that, for each ﬁxed surface, the problem of deciding the embeddability of a given graph is solvable in linear time; however, the multiplicative constant grows rapidly with increasing surface genus. Chapter 4 surveys algorithms for embeddings. Interest in calculating the maximum genus began with Nordhaus, Stewart and White [23]. Although the obvious algorithm based on Xuong’s characterization [45] requires exponential time, Furst, Gross and McGeoch [6] derived a polynomial-time algorithm. Chapter 2 surveys the main results on the maximum genus of a graph. 1 Embedding graphs on surfaces 31

8. Graph minors

There are two natural ways to reduce graphs for the purposes of inductive proofs: deleting an edge (and any isolated vertices that this creates) or contracting an edge, which is defined in a combinatorial context as identifying its endpoints and deleting all resulting loops and multiple edges.Aminor of a graph G is any graph obtainable from G by a sequence of edge deletions and edge contractions. Clearly, if G can be embedded in the surface S, then so can each of its minors. Thus, if G is not embeddable in S, it has a minimal minor that is not embeddable in S,aforbidden minor. As Wagner [42] first observed, Kuratowski’s theorem immediately implies that the forbidden minors for the plane are K5 and K3,3. One of the principal objectives of topological graph theorists for about fifty years was to prove a Kuratowski-type theorem for non-planar surfaces S, that the set of forbidden minors for embeddability in S is finite. This goal was finally attained by Robertson and Seymour [31], and this is described in Chapter 5. Furthermore, they bootstrapped from their results for surfaces to a proof of Wagner’s conjecture that under the partial ordering on all graphs by the minor relation, there are no infinite antichains: in any infinite collection of graphs G1,G2,... , there exist indices i and j with i

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36. W. P. Thurston, Three-Dimensional Topology and Geometry, Princeton Univ. Press, 1997. 37. A. W. Tucker, Branched and folded coverings, Bull. Amer. Math. Soc. 42 (1936), 859–862. 38. T. W. Tucker, The number of groups of a given genus, Trans. Amer. Math. Soc. 258 (1980), 167–179. 39. T. W. Tucker, Groups acting on surfaces and the genus of a group, J. Combin. Theory (B) 34 (1983), 82–98. 40. W. T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21–38. 41. W. T. Tutte, A census of planar maps, Canad. J. Math. 15 (1963), 249–271. 42. K. Wagner, Über eine Eigenschaft der ebene Komplexe, Math. Ann. 114 (1937), 570–590. 43. A. T. White, On the genus of a given group, Trans. Amer. Math. Soc. 173 (1972), 203–214. 44. A. T. White, Graphs of Groups on Surfaces: Interactions and Models, North-Holland, 2001. 45. N. H. Xuong, How to determine the maximum genus of a graph, J. Combin. Theory (B) 26 (1979), 217–225. 2 Maximum genus

JIANER CHEN and YUANQIU HUANG

1. Introduction 2. Characterizations and complexity 3. Kuratowski-type theorems 4. Upper-embeddability 5. Lower bounds References

Since the introductory investigation by Nordhaus, Stewart and White, the maximum genus of a graph has attracted considerable attention from mathematicians and computer scientists. In this survey, we focus on the progress in recent years. In particular, we study its characterizations, algorithmic complexity, upper-embeddability and lower bounds.

1. Introduction

The maximum genus γmax(G) of a graph G is the greatest integer k for which there exists a cellular embedding of G into the orientable surface of genus k. For example, the maximum genus of a tree or a cycle is 0, and the maximum genus of the complete graph K4 is 1. By Euler’s polyhedron formula, if a cellular embedding of a graph G with n vertices, m edges and r faces is on a surface of genus γ , then

n − m + r = 2 − 2γ.

r ≥ γ ≤1 (m − n + ) β(G) = m − n + Since 1, we have 2 1 . The number 1is called the cycle rank of G. It follows that the maximum genus of G is bounded 1 β(G) above by 2 .

34 2 Maximum genus 35

The maximum genus is additive over 2-edge-connected components, in the sense that if e is a cut-edge of a connected graph G and if G1 and G2 are the components of G − e, then γmax(G) = γmax(G1) + γmax(G2) (see [19]). We therefore need to consider only 2-edge-connected graphs. An ear decomposition D =[P1,P2, ··· ,Pr ] of a graph G is a partition of the edge-set of G into an ordered collection of edge-disjoint paths P1, P2, ···, Pr such that P1 is a cycle and, for i ≥ 2, Pi is a path with only its endpoints in common with P1 ∪···∪Pi−1. Each path Pi is called an ear. It is well known (see [24]) that a graph G has an ear decomposition if and only if it is 2-edge-connected. The operations of edge-insertion and edge-deletion have turned out to be useful in the study of graph embeddings. Let ρ(G) be an embedding of a graph G. We say that a new edge e is inserted into ρ(G) if the two endpoints of e are inserted into the corners of faces in ρ(G), yielding an embedding of the graph G + e. If the two endpoints of e are inserted into corners of the same face f in ρ(G), then the edge e ‘splits’ the face f into two faces and leaves the embedding genus unchanged. In this case, the two sides of the new edge e belong to two different faces in the resulting embedding of G + e (see Fig. 1(a)).

(a) (b)

Fig. 1. Inserting an edge

On the other hand, if the two endpoints of e are inserted into corners of two different faces f1 and f2 in ρ(G), then the edge e ‘merges’the faces f1 and f2 into a single larger face and this increases the embedding genus by 1. In this case, the two sides of the new edge e belong to the same face in the resulting embedding of G+e. Topologically, this operation is implemented by cutting along the boundaries of the two faces f1 and f2, leaving two holes on the surface, and then adding a handle to the surface by pasting the two endpoints of an open cylinder to the boundaries of the two holes, so that the new edge e now runs along the new handle (see Fig. 1(b)). The discussion of the edge-deletion operation can be described in the reversed order. Let ρ(G) be an embedding of a graph G, and let e be an edge that is not a cut-edge. If the two sides of e belong to two different faces in ρ(G), then deleting e from ρ(G) ‘merges’ the two faces without changing the embedding genus; if the two sides of e belong to the same face in ρ(G), then deleting e from ρ(G) ‘splits’ the face into two faces and decreases the embedding genus by 1. 36 Jianer Chen and Yuanqiu Huang

2. Characterizations and complexity

It is an interesting problem to characterize the maximum genus of a graph in terms of the combinatorial structures of the graph. This leads to a better and deeper understanding of maximum genus, while a combinatorial characterization may lead to efﬁcient constructions of maximum-genus embeddings. There have been several successful characterizations of maximum genus, and this section gives a summary of them. Let T be a spanning tree of a graph G. The edge complement G − T is called a co-tree; note that the number of edges in any co-tree of G is the cycle rank β(G). A co-tree G − T need not be connected, and a connected component of G − T is even or odd, according to the parity of the number of edges in it. The deﬁciency ξ(G) of a graph G is the minimum number of odd components in any co-tree of G. Any tree whose co-tree achieves this minimum is called a Xuong tree. The following result is due to Xuong [25]:

G γ (G) = 1 (β(G) − ξ(G)) Theorem 2.1 For every graph , max 2 .

The proof of Theorem 2.1 is based on the observation that properly adding two adjacent edges e1 and e2 to an embedding ρ(G) of a graph G can always increase the embedding genus: we first add the edge e1 to ρ(G). If this already increases the embedding genus, then we arbitrarily add the edge e2, noting that adding an edge can never decrease the embedding genus. If inserting e1 does not increase the embedding genus, then e1 must be inserted to the corners of the same face in ρ(G). Thus, the two sides of e1 belong to two different faces of the resulting embedding. Since e2 and e1 share a vertex, we can always insert e2 into the embedding so that the two endpoints of e2 are inserted into corners of two different faces, thus increasing the embedding genus. Now consider a Xuong tree T of the graph G. According to the definition, we 1 (β(G) − ξ(G)) G − T can find 2 pairs of adjacent edges in the co-tree . Thus, if we start with an embedding of the tree T , and add properly the pairs of adjacent edges in G−T to the embedding, we can then obtain an embedding of the graph G whose 1 (β(G) − ξ(G)) γ (G) ≥ 1 (β(G) − ξ(G)) genus is at least 2 . This shows that max 2 . On the other hand, starting with a maximum-genus embedding of the graph G, Chen and Kanchi [6] have shown that a spanning tree T of G can be constructed so that G − T has at least γmax(G) pairs of adjacent edges. Therefore, γmax(G) ≤ 1 (β(G) − ξ(G)) 2 . This theorem also indicates some interesting properties of the maximum genus. Combining Euler’s polyhedron formula and Theorem 2.1, we deduce that the number of faces in a maximum-genus embedding of a graph G is ξ(G) + 1. Moreover, since the cycle rank β(G) is the number of edges in a co-tree, it is easy 2 Maximum genus 37 to see that if β(G) is odd, then ξ(G) ≥ 1 and a maximum-genus embedding of G has at least two faces. Nebeský [17] provided a different characterization. For a set A of edges of G, let Co(G − A) and Ce(G − A) be the numbers of components in G − A with odd and even cycle rank, respectively, and let

ν(G,A) = Ce(G − A) + 2Co(G − A) −|A|−1.

Finally, let ν(G) denote the maximum value of ν(G,A), taken over all subsets A of the edge-set E(G).

Theorem 2.2 For every graph G, ν(G) = ξ(G).

Combining Theorem 2.1 and Theorem 2.2, we immediately get another γ (G) = 1 (β(G) − ν(G)) characterization of the maximum genus: max 2 . We outline a proof for one direction of Theorem 2.2. Let ρ(G) be a maximum- genus embedding of the graph; then the number of faces in ρ(G) is ξ(G) + 1. Let A be a set of edges, and consider the process of successively removing the edges of A from ρ(G). Since removing an edge from an embedding increases the number of faces by at most 1, the resulting embedding of G−A has at most ξ(G)+|A|+1 faces. But the embedding of G−A is a collection of embeddings of the components of G−A.An embedding of a component with even cycle rank has at least one face, while an embedding of a component with odd cycle rank has at least two. Thus, the total number of faces in the embedding of G−A is at least Ce(G, A)+2Co(G, A), and so ξ(G)+|A|+1 ≥ Ce(G, A)+2Co(G, A), or equivalently, ξ(G) ≥ ν(G,A). Since A is an arbitrary set of edges in G, ξ(G) ≥ ν(G). Nebeský [17] proved that there is an edge-set A in a maximum-genus embedding of the graph G that reverses the above process, thus proving that ξ(G) ≤ ν(G). The maximum genus of a graph can also be characterized in terms of ear decompositions of the graph (see Chen and Kanchi [6]). The computational complexity of finding the maximum genus has an interesting history. A straightforward method is to enumerate all embeddings of a graph and select one with the largest genus. This algorithm is not feasible since the number of embeddings of a graph with n vertices can be as large as ((n − 2)!)n (see [11]). The results of Theorems 2.1 and 2.2 have suggested alternative methods that involve picking a spanning tree or a set of edges. However, the number of spanning trees and the number of edge-sets in a graph with n vertices can still be exponential in n, and no polynomial-time algorithm was found in the immediate years after Theorems 2.1 and 2.2 were published. Eventually, based on the construction of Theorem 2.1, Furst, Gross and McGeoch [8] developed a polynomial-time algorithm that constructs a maximum-genus embedding of a 38 Jianer Chen and Yuanqiu Huang graph by reducing the problem to the linear matroid parity problem, which was known to be solvable in polynomial time (see [9]). For any fixed positive integer k, there is a linear-time algorithm that decides whether a given graph has maximum genus k and, if so, finds an embedding that achieves it (see Chen [2]). Moreover, the isomorphism of two graphs of bounded maximum genus can be tested in linear time [2].

3. Kuratowski-type theorems

Because of Kuratowski’s theorem, a characterization of graph embeddability in terms of a ﬁnite set of graphs has been called a ‘Kuratowski-type’ result. This direction of research has led to the recent exciting developments of the Graph Minor Theory (see Chapter 5). In this section, we develop Kuratowski-type theorems for maximum genus, starting with graphs of maximum genus 0. We note ﬁrst that a 2-edge-connected graph G has maximum genus 0 if and only if G is a cycle. To see why, consider an ear decomposition P1,P2, ···,Pr of G.IfG is not a cycle, then r>1. Now it is easy to construct an embedding of genus 1 for the subgraph P1 ∪ P2. Arbitrarily adding the rest of the ears results in an embedding with genus at least 1 for the graph G. Now let G be a graph of maximum genus 0. If G is not 2-edge-connected, then it contains at least one cut-edge. Removing this cut-edge results in two smaller graphs of maximum genus 0. Based on this observation and using induction on the graph size, we can derive the following theorem:

Theorem 3.1 A graph G has maximum genus 0 if and only if no vertex is on two different cycles of G.

The class of graphs of maximum genus 0 was first characterized by Nordhaus, Stewart and White [19]. They defined a cactus to be a graph that can be constructed from a tree T using the following operation: pick a subset S of vertices in T and replace each vertex in S by a cycle. They proved that a graph G has maximum genus 0 if and only if G is a cactus. It is not difficult to see that this result is equivalent to Theorem 3.1. Next we consider the class of graphs of maximum genus 1. We first discuss the problem for 2-edge-connected graphs. Gross, Klein and Rieper [10] defined a class of graphs called necklaces. Starting with a cycle C2r+s, we obtain a necklace of type (r, s) by doubling r non-adjacent edges in C2r+s and adding a loop at each of the remaining s vertices; the added edges are called beads of the necklace. Fig. 2 shows the necklaces of types (4, 0) and (1, 3). It is easily seen from Theorem 2.1 that the maximum genus of any necklace is 1. 2 Maximum genus 39

Fig. 2. Necklaces of types (4, 0) and (1, 3)

Starting from an ear decomposition of a 2-edge-connected graph and considering all possible attachments of ears, while keeping the maximum genus 1, we can obtain the following result of Chen and Gross [4]:

Theorem 3.2 A 2-edge-connected graph G has maximum genus 1 if and only if it is homeomorphic to a necklace or to one of the graphs in Fig. 3.

Fig. 3.

Finally, we consider general graphs of maximum genus 1. Recall that the maximum genus of a graph is equal to the sum of the maximum genera of its 2-edge-connected components. This fact, combined with the discussion on maximum genus of 2-edge-connected graphs in this section, gives the following result:

Theorem 3.3 A graph G has maximum genus 1 if and only if each of its 2-edge- connected components except one is a cycle or a single vertex, and that one is homeomorphic to a necklace or to one of the graphs in Fig. 3.

4. Upper-embeddability

We have seen that the maximum genus of a graph G cannot be larger than half the cycle rank β(G). It is interesting to characterize those graphs that attain this bound. G 1 β(G) We say that a graph is upper-embeddable if its maximum genus is 2 . For example, consider the graphs in Theorem 3.2. By Theorem 3.2, the maximum genus of these graphs is 1. On the other hand, it is easy to verify that the cycle rank of these graphs is either 2 or 3, and so all of these graphs are upper-embeddable. Early research on upper-embeddability focused on identifying classes of graphs that are upper-embeddable; for example, all complete graphs and complete 40 Jianer Chen and Yuanqiu Huang bipartite graphs are upper-embeddable. Since the publication of Theorems 2.1 and 2.2, researchers have become more interested in the study of combinatorial structures that make a graph upper-embeddable. In particular, by Theorem 2.1, we γ (G) = 1 (β(G) − ξ(G)) β(G) ξ(G) have max 2 . Since and have the same parity, we obtain the following result: Theorem 4.1 A graph G is upper-embeddable if and only if its deﬁciency ξ(G) is 0 or 1 — that is, G has a spanning tree T for which the co-tree G − T has at most one odd component. Re-interpreting Theorem 4.1 using the notation in Theorem 2.2, we deduce that a graph G is upper-embeddable if and only if

Ce(G − A) + 2Co(G − A) − 2 ≤|A|, for each set A of edges in G.

Theorem 4.1 was discovered independently by a number of researchers (see [14], [17], [19] and [26]). It has turned out to be very powerful for identifying upper-embeddable graphs. For example, if a graph has a spanning tree that has a connected co-tree, then the conditions in Theorem 4.1 are trivially satisfied and the graph is upper-embeddable. Since every 4-edge-connected graph contains two edge-disjoint spanning trees (see Kundu [15]), we immediately deduce the following result: Corollary 4.2 Every 4-edge-connected graph is upper-embeddable. This result cannot be extended to 3-edge-connected graphs. Chen, Kanchi and Gross [7] gave a general construction of such graphs that are not upper- embeddable. Consequently, the study of upper-embeddability must focus on graphs with edge-connectivity at most 3. A number of upper-embeddable graph classes have been identified, based on Theorems 2.1 and 2.2, including locally connected graphs [18], cyclically 4-edge- connected graphs [20], k-regular vertex-transitive graphs of girth g with k ≥ 4or g ≥ 4 [23], loop-free graphs of diameter 2 [22], (4k + 2)-regular graphs and 2k- regular bipartite graphs [12]; see the original papers for more detailed definitions and results. Finally, we note that, since the maximum genus of a graph can be determined in polynomial time, upper-embeddability can also be tested in polynomial time.

5. Lower bounds

Upper-embeddability of a graph gives a lower bound on the maximum genus in terms of the cycle rank of the graph. Therefore, the maximum genus of a 4-edge- G 1 β(G) connected graph is at least (and in fact equals) 2 . More recently there 2 Maximum genus 41 has been much interest in deriving lower bounds for the maximum genus of graphs that are not upper-embeddable. In general, there is no lower bound for the maximum genus in terms of graph size (that is, the number of vertices or the number of edges) or of cycle rank. For example, necklaces can have an arbitrarily large size and cycle rank, but a maximum genus bounded by 1 (Theorem 3.2). Therefore, in order to derive meaningful lower bounds for the graph maximum genus, we must exclude necklaces. Observe that a necklace contains many disjoint loops and multiple edges. It can be shown (see [5]) that this is essentially the only way to construct graphs of low maximum genus but large cycle rank. Moreover, if vertices of degree 1 and 2 are allowed, then we can easily eliminate the necklace structures without changing the cycle rank, by subdividing edges in the graph by vertices of degree 2 or by introducing new edges with one endpoint of degree 1 and the other endpoint subdividing an edge in the graph. This leads us to the study of lower bounds on the maximum genus for simple graphs with a minimum degree of at least 3. The following result is due to Chen, Kanchi and Gross [7]: Theorem 5.1 If G is a simple graph with minimum degree at least 3, then γ (G) ≥1 β(G) max 4 . Theorem 5.1 was proved, using Theorem 2.1, by carefully counting and comparing the numbers of odd and even components in the co-tree of a Xuong tree. We note that it is sufﬁcient to prove the theorem for 3-regular simple graphs.A vertex-splitting operation then easily generalizes the theorem to all simple graphs with minimum degree at least 3 (see [7]). The bound in Theorem 5.1 is sharp, in the sense that there are simple graphs G 1 β(G) with minimum degree 3 whose maximum genus is arbitrarily close to 4 ; Fig. 4 illustrates a general construction of such a graph. In general, such a graph is constructed by connecting each bead in a type (d, 0) necklace to a separate copy of K4 by a cut-edge. It is not difﬁcult to verify that the ratio of the maximum genus to the cycle rank of this graph is (d + 1)/(4d + 1), which can be arbitrarily 1 close to 4 .

1 β(G) Fig. 4. Simple graphs with maximum genus close to 4 42 Jianer Chen and Yuanqiu Huang

Observe that the graphs constructed in Fig. 4 contain many cut-edges. One can ask whether the lower bound in Theorem 5.1 can be further improved if a graph is 2-edge-connected. This is indeed the case, as shown by the following theorem.

Theorem 5.2 If G is a 2-edge-connected simple graph with minimum degree at γ (G) ≥1 β(G) least 3, then max 3 . This theorem was first proved by Kanchi and Chen usingTheorem 2.1.Archdeacon, Nedela and Skoviera˘ gave a simpler proof based on Theorem 2.2. The results have been combined and reported in [1]. Comparing Theorems 5.1 and 5.2, we may ask whether the lower bound can be further improved for 3-edge-connected graphs. That it cannot was shown by Chen, Archdeacon and Gross [3], who exhibited an infinite class of 3-edge-connected G 1 β(G) simple graphs with maximum genus 3 . On the other hand, 3-edge-connectivity automatically excludes vertices of degree less than 3 and the necklace structure. Thus, we might ask whether, without the constraint of being a simple graph, there is a non-trivial lower bound on the maximum genus for 3-edge-connected graphs. This conjecture was confirmed independently, around the same time, by at least four research groups, and their results are reported in [1]. G γ (G) ≥1 β(G) Theorem 5.3 If is 3-edge-connected, then max 3 . The same construction in [3] shows that the bounds in Theorems 5.2 and 5.3 are sharp. Since k-vertex-connectivity implies k-edge-connectivity, the results described in this section for k-edge-connected graphs can be translated immediately to k- vertex-connected graphs [3]. We summarize the results in Table 1, where we have also included the results of Corollary 4.2. Here, ‘infinitely tight’ means that there are infinitely many graphs in the corresponding class that achieve the bound. In particular, this column indicates that for each of the graph classes, there are infinitely large graphs in the class that achieve the corresponding bound.

Table 1.

kγmax(G) inﬁnitely tight? remark ≥1 β(G) 1 4 yes true only for simple graphs ≥1 β(G) 2 3 yes true only for simple graphs ≥1 β(G) 3 3 yes true for general graphs ≥ =1 β(G) 4 2 yes true for general graphs 2 Maximum genus 43

Finally, we note that recent research has studied lower bounds on the maximum genus in terms of other parameters, including connectivity, independence number, girth and chromatic number (see [13] and [16]).

References

1. D.Archdeacon, J. Chen,Y.Huang, S. P.Kanchi, D. Li,Y.Liu, R. Nedela and M. Skoviera,˘ Maximum genus, connectivity, and Nebeský’s theorem, Discrete Math., to appear. 2. J. Chen, A linear time algorithm for isomorphism of graphs of bounded average genus, SIAM J. Discrete Math. 7 (1994), 614–631. 3. J. Chen, D. Archdeacon and J. L. Gross, Maximum genus and connectivity, Discrete Math. 149 (1996), 19–29. 4. J. Chen and J. L. Gross, Kuratowski-type theorems for average genus, J. Combin. Theory (B) 57 (1993), 100–121. 5. J. Chen and J. L. Gross, No lower limit points for average genus, Graph Theory, Combinatorics, and Applications (eds. Y. Alavi and A. Schwenk), Wiley Interscience (1995), 183–194. 6. J. Chen and S. P. Kanchi, Graph ear decompositions and graph embeddings, SIAM J. Discrete Math. 12 (1999), 229–242. 7. J. Chen, S. P. Kanchi and J. L. Gross, A tight lower bound on the maximum genus of a simplicial graph, Discrete Math. 156 (1996), 83–102. 8. M. Furst, J. L. Gross and L. A. McGeoch, Finding a maximum-genus graph imbedding, J. Assoc. Comput. Mach. 35 (1988), 523–534. 9. H. N. Gabow and M. Stallmann, Efﬁcient algorithms for graphic matroid intersection and parity, Lecture Notes in Computer Science 194 (1985), 210–220. 10. J. L. Gross, E. W. Klein and R. G. Rieper, On the average genus of a graph, Graphs and Combinatorics 9 (1993), 153–162. 11. J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, 1987 and Dover, 2001. 12. Y. Huang and Y. Liu, The classes of upper embeddable graphs with the same value of degree of vertex under modulo, Acta Math. Sci. 20 (2000), 251–255. 13. Y.Huang and Y.Liu, Maximum genus, independence number and girth, Chinese Annals of Math. 21 (2000), 77–82. 14. M. Jungerman,Acharacterization of upper-embeddable graphs, Trans.Amer. Math. Soc. 241 (1978), 401–406. 15. S. Kundu, Bounds on the number of disjoint spanning trees, J. Combin. Theory (B) 17 (1974), 199–203. 16. D. Li and Y. Liu, Maximum genus, girth and connectivity, European J. Combin. 21 (2000), 651–657. 17. L. Nebeský, A new characterization of the maximum genus of a graph, Czech. Math. J. 31 (1981), 604–613. 18. L. Nebeský, Every connected, locally connected graph is upper embeddable, J. Graph Theory 5 (1981), 205–207. 19. E. Nordhaus, B. Stewart and A. White, On the maximum genus of a graph, J. Combin. Theory (B) 11 (1971), 258–267. 20. C. Payan and N. H. Xuong, Upper embeddability and connectivity of graphs, Discrete Math. 27 (1979), 71–80. 44 Jianer Chen and Yuanqiu Huang

21. R. Ringeisen, Survey of results on the maximum genus of a graph, J. Graph Theory 3 (1979), 1–13. 22. M. Skoviera,˘ The maximum genus of graphs of diameter two, Discrete Math. 87 (1991), 175–180. 23. M. Skoviera˘ and R. Nedela, The maximum genus of vertex-transitive graphs, Discrete Math. 78 (1989), 179–186. 24. H. Whitney, Non-separable and planar graphs, Trans. Amer. Math. Soc. 34 (1932), 339–362. 25. N. H. Xuong, How to determine the maximum genus of a graph, J. Combin. Theory (B) 26 (1979), 217–225. 26. N. H. Xuong, Upper-embeddable graphs and related topics, J. Combin. Theory (B) 26 (1979), 226–232. 3 Distribution of embeddings

JONATHAN L. GROSS

1. Introduction 2. Enumerating embeddings by surface type 3. Total embedding distributions 4. Congruence classes 5. The unimodality problem 6. Average genus 7. Stratiﬁcation of embeddings References

The principal genus distribution problem is to count the number of cellular embeddings of a given graph. Complete distributions have been obtained for a few basic families of graphs. Various properties of genus distributions and of related invariants are examined, especially the properties of the average genus.

1. Introduction

A ubiquitous question in topological graph theory is whether a given graph can be embedded in a given surface, a question that readily extends to the problem of counting the number of different embeddings of that graph into that surface. (A contrasting classical problem with its origins in geometry asks, for a ﬁxed surface, how many different maps there are onto that surface, where what varies is the graph that serves as the 1-skeleton.) This chapter explores the programme introduced by Gross and Furst [11] of constructing surface-by-surface inventories of the embeddings of a ﬁxed graph into not just one surface, but every surface, and gives the related theory. All embeddings are taken to be cellular, except where it is clear from context that non-cellular embeddings are under consideration.Two cellular embeddings are

45 46 Jonathan L. Gross considered to be the same if their rotation systems are combinatorially equivalent (see Chapter 1). Given a graph G and an orientable surface Sh, the number of embeddings of G in Sh is the number gh(G) of rotation systems for G that induce a cellular embedding in Sh. The orientable genus range of a graph G is the set of integers h for which gh(G) > 0. The minimum genus and maximum genus are the numbers

γmin(G) = min{h: gh(G) > 0} and γmax(G) = max{h: gh(G) > 0}.

The following result is a variation on an elementary theorem that ﬁrst appeared in [8] and is often called an ‘intermediate value theorem for genus’. The proof proceeds by showing that we can change any rotation system of a graph into any other by effecting a sequence of transpositions of edge-ends, each of which changes the genus of the induced surface by at most 1. In particular, there is such a sequence that goes from a minimum-genus embedding to a maximum-genus embedding. Theorem 1.1 The orientable genus range of a graph G is the consecutive set of integers {h: γmin(G) ≤ h ≤ γmax(G)}.

The genus distribution of a graph G is the integer-valued function h → gh(G), and the genus distribution polynomial of G is h IG(x) = gh(G) x . h≥0

These concepts were introduced by Gross and Furst [11]. Theorem 1.2 For any graph G, gh(G) = (deg(v) − 1)!. h≥0 v∈V (G)

Proof The sum on the left and the product on the right both count each embedding of G exactly once. For relatively small graphs, the orientable genus distribution can be calculated by elementary ad hoc methods. For example, in Fig. 1, the graph K2 2 C3 has six vertices, each of degree 3, and so the total number of orientable embeddings is 64 = 26. A bar-amalgamation of two disjoint graphs G and H is obtained from G ∪ H by joining a vertex of G and a vertex of H with an edge, referred to as the bar,as illustrated in Fig. 2. The following theorem of Gross and Furst [11] is useful in simplifying genus distribution calculations. 3 Distribution of embeddings 47

Genus of surface 01 2

K2 C3 Number of embeddings 2 38 24

Fig. 1. A graph and its orientable genus distribution

Fig. 2. A bar-amalgamation of K4 and K5 − e

Theorem 1.3 The genus distribution of a bar-amalgamation of two graphs is the convolution of their respective genus distributions, multiplied by the product of the degrees of the two vertices of the bar (not counting the bar).

The outcome is very much more complicated when two graphs are directly amalgamated at a vertex, without the bar. Of course, two graphs can be amalgamated along an edge in each graph or along any pair of isomorphic subgraphs. The number of embeddings g˜k(G) of a graph G in the non-orientable surface Nk is the number of general rotation systems for G that induce an embedding in Nk. Analogous to the genus range for the orientable case, the crosscap range of G is the set of integers {k :˜gk(G) > 0}. The numbers

γ˜min(G) = min{k :˜gk(G) > 0} and γ˜max(G) = max{k :˜gk(G) > 0} are called the minimum crosscap number and maximum crosscap number. The crosscap distribution (or non-orientable genus distribution) of a graph G is the integer-valued function k →˜gk(G), and the crosscap distribution polynomial is k I˜G(y) = g˜k(G) y . k≥1

The following two basic theorems for the embeddings of a graph in non-orientable surfaces are analogous to the results for orientable surfaces. 48 Jonathan L. Gross

Theorem 1.4 The crosscap range of a graph G is the set

{k :˜gmin(G) ≤ k ≤˜gmax(G)}.

Proof Twisting an edge in a general rotation system changes the Euler characteristic of the resulting induced surface by at most 1. Transposing two edge- ends changes the Euler characteristic of the resulting induced surface by at most 2. It is possible to change any general rotation system into any other by effecting a sequence of operations, each either an edge-twist or a transposition of edge-ends. In particular, the intermediate crosscap number values between the minimum and maximum are all realizable. The details are slightly more intricate than for Theorem 1.3. Theorem 1.5 Let G be a graph and let β(G) be its cycle rank. Then β(G) gh(G) + g˜k(G) = 2 (deg(v) − 1)!. h≥0 k≥1 v∈V (G)

Proof The sum on the left and the product on the right both count every embedding of the graph G exactly once. The factor of 2β(G) on the right accounts for the number of possible ways to orient the edges that are not in a speciﬁed spanning tree of G.

2. Enumerating embeddings by surface type

Even at the outset of the programme to provide explicit calculations of embedding distributions, it was clear that a variety of techniques would be needed. In this section, we consider three inﬁnite families: closed-end ladders, cobblestone paths and bouquets.

Closed-end ladders

The closed-end ladder Ln is the graph obtained from the Cartesian product Pn 2 K2 by doubling the edges at the ends of the path, as illustrated in Fig. 3. Such ladder-like graphs played a crucial role in the solution of the Heawood map-colouring problem (see Ringel [27]).

Fig. 3. The 3-rung closed-end ladder L3 3 Distribution of embeddings 49

A topological lemma based on a face-tracing argument was used by Furst, Gross and Statman [10] to partition the possible rotation systems of the ladder Ln according to the induced surface genus. This permitted the number gh(Ln) to be represented as a sum whose terms are products of binomial coefﬁcients. A succession of combinatorial identities led to the following closed formula:   n− +h n + 1 − h 2n + 2 − 3h 2 1 for h ≤ 1 (n + 1) , g (L ) = h n + 1 − h 2 h n  0 otherwise.

Table 1 shows the genus distribution for some small ladders.

Table 1. The genus distribution of small ladders

gh(Ln)g0 g1 g2 g3 g4 total

L1 220004 L2 41200016 L3 8401600 64 L4 16 112 128 0 0 256 L5 32 288 576 128 0 1024

Subsequently, McGeoch [23] calculated the genus distributions of the related families of graphs known as circular ladders and Möbius ladders. Also, Tesar [31] calculated the genus distribution of the family known as Ringel ladders.

Cobblestone paths

The cobblestone path Jn is the graph obtained by doubling each edge of the n-vertex path Pn and then adding a loop at each end, as illustrated in Fig. 4.

Fig. 4. The cobblestone path J3

The genus distribution of cobblestone paths was calculated recursively by Furst, Gross and Statman [10], starting from J0. They derived a pair of simultaneous recursions by using topological considerations, in which the right-most loop is subdivided, and a new loop is joined to the new vertex. They simpliﬁed the simultaneous recursion to a single recursion and solved it with the aid of generating 50 Jonathan L. Gross functions into the following closed form, for h ≥ 0 and n ≥ 1: n − h n − h h n−1 h−1 n−1 gh(Jn) = 3 · 4 + 2 · 3 · 4 . h h − 1

Table 2 shows the genus distribution for some of the smaller cobblestone paths.

Table 2. The genus distribution of some cobblestone paths

gh(Jn)g0 g1 g2 g3 total

J1 4200 6 J2 16 20 0 0 36 J3 64 128 24 0 216 J4 256 704 336 0 1296

These early computations of embedding distributions for ladders and cobblestone paths were later elaborated upon by Stahl [29], who introduced the idea of ‘linear families’. In [30] he also studied embedding distributions for some graphs of small diameter.

Bouquets

The bouquet B is the graph with one vertex and loops, as illustrated in Fig. 5. One of the properties of bouquets important to topological graph theory is that every regular graph can be derived by assigning voltages (possibly permutation voltages) to some bouquet (see Gross and Tucker [15], [17]).

B1 B2 B3

Fig. 5. Some small bouquets

Permutation-group algebra is a key to calculating the distribution of embeddings of bouquets. Gross, Robbins and Tucker [14] established the equation

n−1 gh(Bn) = (n − 1)!·2 · en−2h+1(n).

The quantity ek() is the cardinality of the set of permutations π ∈ 2, corresponding to an arbitrary ﬁxed cycle ζ of length 2 for which there is a full involution β such that π = ζ ◦ β, and such that π has k cycles. The value of ek() 3 Distribution of embeddings 51

is given by a formula of Jackson [19]. The closed formula above for gh(B) leads to the following recursion: initial conditions:

g0(B0) = 1,g0(B1) = 1 and gh(B0) = gh(B1) = 0 for h ≥ 1; g0(B2) = 4,g1(B2) = 2 and gh(B2) = 0, for h ≥ 2. recursion for h>2: 2 ( + 1)gh(B) = 4(2 − 1)(2 − 3)( − 1) ( − 2)gh−1(B−2) + 4(2 − 1)( − 1)gh(B−1). This recursion enables us to calculate the numerical values for the genus distribution of bouquets in Table 3.

Table 3. The genus distribution of some bouquets

gh(Jn)g0 g1 g2 total

B0 1001 B1 1001! B2 4203! B3 40 80 0 5! B4 672 3360 1008 7! B5 16128 161280 185472 9!

Rieper [26] extended the group-character approach of [14] in his analysis of the genus distribution of dipoles. (The dipole D is the graph with two vertices joined by edges.)

3. Total embedding distributions

When non-orientable embeddings of a graph G are also included, the total number of embeddings increases by a factor of 2β(G)−1 since, for any given rotation system and ﬁxed spanning tree T , each of the β(G) edges not in T may be twisted or untwisted. The total embedding distribution of a graph G is the bivariate polynomial h k I¨G(x, y) = IG(x) + I˜G(y) = gh(G)x + g˜k(G)y . h≥0 k≥1

The calculation of total embedding distributions appears to be quite difﬁcult, in part because the possible twisting of edges complicates the recurrences that one might derive. In order to calculate the genus or crosscap number for a given general 52 Jonathan L. Gross rotation system ρ without doing face-tracing, we can choose a spanning tree T , and then calculate the entries of the overlap matrix Mρ,T = (mi,j ), in which   i = j (ρ) , 1if and pure T +ei +ej is non-planar m = − i = j i , i,j  1if and edge is twisted  0 otherwise.

Here, the notation pure(ρ)|T +ei +ej means the restriction of the underlying pure part of the rotation system ρ to the subgraph T + ei + ej . Mohar [22] derived the following general property of the overlap matrix. Theorem 3.1 Let G be a graph, let T be a spanning tree of G, and let ρ be a general rotation system of G. Then 2h if the induced surface S(ρ) is Sh, rank(Mρ,T ) = k if the induced surface S(ρ) is Nk.

For example, in Fig. 6, the edges of the spanning tree are labelled 4, 5 and 6. Thus, the rows and columns correspond to the co-tree edges 1, 2 and 3. The cross on the bottom edge indicates that it is twisted. Since the rank of the overlap matrix is 3 and the embedding is non-orientable, the surface for that embedding must be N3.

001 4 6 010 5 100 overlap 1 3 matrix x 2

Fig. 6. A graph and its overlap matrix

At ﬁrst, the concept of an overlap matrix seemed to be purely of theoretical interest. Indeed, whereas calculating the surface type by face-tracing requires O(m) time for a graph with m edges, calculating the rank of an overlap matrix deteriorates to O(m2) time. However, Chen, Gross and Rieper [6] discovered that regrouping the total set of embeddings according to the rank of the overlap matrix sometimes facilitates the calculation of the total embedding distribution. For example, consider a tree T in the ladder graph for which the co-tree is a path, as in Fig. 7, and in the cobblestone path graph for which a co-tree is almost a path, as in Fig. 8. These yield a ‘tridiagonal’ overlap matrix, which is convenient 3 Distribution of embeddings 53

1 7

234 5 6

Fig. 7. The ladder L6 with a spanning tree, and the form of the corresponding tridiagonal overlap matrix

Fig. 8. The cobblestone path J5 with a spanning tree for calculating the rank because there are zeros everywhere except possibly on a narrow band near the diagonal. Each (x) in the matrix of Fig. 7 indicates an entry that may be either 1 or 0. The following total embedding distribution polynomial for closed-end ladders was obtained in [6]: r ih ih+1 n n+1−r 2 2 2 2 y rd + rd y − IL (y ) + IL (x), 3 3 n n h=1 where rd(x) means the nearest integer to x, and the sum is taken over all r-tuples of positive integers i1,i2, ..., ir with sum equal to n + 1. This total embedding distribution polynomial was obtained for cobblestone paths:

r ih ih+1 n+r−1 n+1−r 2 2 2 2 y rd + rd y − IJ (y ) + IJ (x). 3 3 n n h=1

4. Congruence classes

A second enumerative aspect of graph embeddings regards two embeddings as equivalent if they ‘look alike’ when vertex and edge labels are removed. Two embeddings ι1: G → S and ι2: G → S are congruent, denoted by ι1 ι2, if there exist a graph automorphism α: G → G and a surface homeomorphism η: S → S for which the diagram in Fig. 9 is commutative. 54 Jonathan L. Gross

ι1 G S α η

G S ι 2

Fig. 9. The commutativity condition for an embedding congruence

For example, Fig. 10 shows how the sixteen different orientable embeddings of the complete graph K4 are partitioned into congruence classes.

two like this in S0 six like this in S1 eight like this in S2

Fig. 10. Partitioning the embeddings of K4 into congruence classes

Burnside’s lemma is used to count congruence classes. Each automorphism of a graph G induces a permutation on the rotation systems of G that preserves the congruence class, but does not necessarily preserve the equivalence class, as illustrated in Fig. 11.

1 1.234 4 2.134 2 3. 1 2 4 3 4. 1 2 3

(1 2 3)(4) (1 3)(2 4) changes fixes

2. 3 1 4 3 3 3.412 3.214 4 2 4.312 1.234 1.342 1 4 4. 2 3 1 2.341 2 1

Fig. 11. The induced action of two permutations on a rotation system Fig. 11 indicates the induced action on a given rotation system of the permuations (123)(4) and (13)(24). Although rows 1 and 4 of the resulting 3 Distribution of embeddings 55 rotation system for (123)(4) (lower left) are the same rotations as in the given rotation system (top), the rows 2 and 3 are the reverse rotations. By way of contrast, rows 1, 2, 3 and 4 of the resulting rotation system for the permutation (13)(24) (lower right) are the same as the corresponding rows of the given permutation. The key to counting congruence classes has been to convert the cycle index of Aut(G) acting on VG into the cycle index for the induced action on the rotation systems. Mull, Rieper and White [24] showed how to count congruence classes of embeddings of complete graphs into oriented surfaces. A subsequent generalization by Kwak and Lee [21] can also be used to count congruence classes of non-orientable embeddings. One of their underlying ideas is to regard an edge-twist as having voltage 1 (mod 2) and to construct the orientable double cover. Then the graph automorphisms act on the induced rotation systems.

5. The unimodality problem

A sequence {am} is unimodal if there exists an integer M such that

am−1 ≤ am for all m ≤ M and am ≥ am+1 for all m ≥ M.

A typical unimodal sequence ﬁrst rises and then falls. A sequence {am} is strongly unimodal if its convolution with every unimodal sequence yields a unimodal sequence. An equivalent criterion for unimodality (see [20]) is that

2 am ≥ am+1am−1, for all m.

It is easily proved that every strongly unimodal sequence is unimodal, which afﬁrms the appropriateness of the name ‘strongly unimodal’. All known genus distributions are strongly unimodal. The pioneering calculations are summarized in the next theorem, for which the ﬁrst two families were studied by Furst, Gross and Statman [10] and the third by Gross, Robbins and Tucker [14]. Theorem 5.1 The genus distributions of closed-end ladder graphs, cobblestone paths and bouquets are strongly unimodal.

We observe that an embedding of the bouquet B has + 1 faces in the sphere S0, − 1 faces in the torus S1, − 3 faces in the double-torus S2, and so on. This suggests that the genus distribution of the bouquet Bn might resemble this sequence of Stirling cycle numbers n n n 2 , 2 , 2 , ··· , n + 1 n − 1 n − 3 n where k denotes the number of ways to partition n distinct objects into k cycles. 56 Jonathan L. Gross

Using group character theory, Stahl [28] proved that the genus distribution of bouquets is asymptotically proportional to this sequence. He also proved [30] that the resemblance to Stirling numbers also holds for various graphs of small diameter, including partial suspensions of trees and of cycles, which serves as further evidence for unimodality. Whether the genus distribution of every graph is strongly unimodal remains an interesting open problem.

6. Average Genus

The average value of the genus of the embedding surface for a graph G, taken over all orientable embeddings, is called the average genus and denoted by γavg(G). Obviously, a graph has average genus 0 if and only if it has maximum genus 0. Thus, in consideration of Nordhaus et al. [25], a graph has average genus 0 if and only if no vertex lies on more than one cycle. As a corollary to Theorem 1.3, Gross and Furst [11] proved that the average genus acts additively on bar-amalgamations. (Analogously, both γmin and γmax act additively on bar-amalgamations.) Theorem 6.1 The average genus of a bar-amalgamation of two graphs G and H is γavg(G) + γavg(H ). Suppose that r independent edges of a 2r + s cycle are doubled and that a loop is added at each vertex not on a doubled edge, as illustrated in Fig. 12. Such a graph is called a necklace of type (r, s). Gross, Klein and Rieper [12] calculated the average genus of all such graphs.

Fig. 12. A necklace of type (2, 4)

r s (r, s) − 1 2 . Theorem 6.2 The average genus of a necklace of type is 1 2 3 Since the number of different necklaces of type (r, s) grows arbitrarily large as the numbers r and s increase, it follows that arbitrarily many non-homeomorphic 2-connected graphs can have the same average genus. By Xuong’s theorem (see Chapter 2), the maximum genus of a necklace is 1. If r or s increases, then the 3 Distribution of embeddings 57 average genus of a necklace of type (r, s) approaches 1, which is part of the general pattern that a randomly chosen embedding is more likely to be nearer to the maximum genus than to the minimum genus. This theorem also established the number 1 as the ﬁrst known upper limit point of average genus, thereby raising the questions of the prevalence of upper limit points and of the existence of lower limit points. These are discussed in greater detail below. 1 1 5 2 19 The six smallest values of the average genus of any graph are 3 , 2 , 9 , 3 , 27 3 and 4 (see [12]). Fig. 13 shows a necklace realizing each of them.

1 1 5 2 19 3 3 2 9 3 27 4

Fig. 13. The graphs with the six smallest positive values of the average genus

Chen and Gross [4] found that, except for necklaces, there are just eight 2-connected graphs of average genus less than 1. Their proof is based on the minimum number of paths that one must add to a cycle to obtain a given 2-connected graph — that is, on the number of steps in a ‘Whitney synthesis’(also called an ‘ear decomposition’). The bouquet B3, the dipole D4 and the complete K 2 5 7 graph 4 have average genus 3 , 6 and 8 , respectively. Fig. 14 shows the other ﬁve such graphs and their average genus. In combination with Theorem 1.3, this yields a complete classiﬁcation of those graphs with average genus less than 1.

5 8 3 17 8 6 9 4 18 9

Fig. 14. The ﬁve sporadic 2-connected graphs with average genus less than 1

Further, it is proved in [4] that there are exactly three 2-connected graphs with average genus exactly 1. They are shown in Fig. 15. Theorem 6.2 inspired an investigation of the possible limit points of average genus, and Chen and Gross [2] obtained the following results for 3-connected graphs (simple or not) and for simple 2-connected graphs. 58 Jonathan L. Gross

Fig. 15. The three 2-connected graphs with average genus 1

Theorem 6.3 For each real number r, there are only ﬁnitely many 3-connected graphs and ﬁnitely many simple 2-connected graphs with average genus less than r.

Consequently, there are only ﬁnitely many 3-connected or simple 2-connected graphs with the same average genus. Moreover, the sets of values of the average genus of these large families of graphs have no limit points. In a sequel, Chen and Gross [3] gave a systematic method for constructing upper limit points, which we now describe. Let e be an edge of a graph. We attach an open ear to an edge e if we insert two new vertices in the interior of e and then double the edge between them; the two new vertices are called the ends of that open ear. Similarly, we attach a closed ear to the edge e if we insert one new vertex w in the interior of e and then attach a loop at w; the vertex w is called the end of that closed ear. Fig. 16 illustrates both kinds of ear attachments.

closed ears

open ears

Fig. 16. Three open ears and two closed ears serially attached to an edge e

We say that r open ears and s closed ears are attached serially to the edge e if the ends of the ears are all distinct, and if no ear has an end between the two ends of an open ear. Theorem 6.4 provides a means for constructing upper limit points; in fact, all limit points arise from this construction. 3 Distribution of embeddings 59

Theorem 6.4 Let G be a 2-connected graph, and let G+ be a graph obtained by serially attaching ears to an edge of G. Then

γavg(G)<γavg(G+)<γavg(G) + 1.

Chen and Gross [5] concluded their series of papers by proving that the set of values of average genus taken over all graphs has no lower limit points. A linear- time algorithm for isomorphism testing of graphs of bounded average genus was derived by Chen [1]. Some bounds have been obtained for the average genus. In this summary of what is known about bounds, it has been conjectured that part (c) can be improved. In the following theorem, part (a) is due to Gross, Klein and Rieper [12] and parts (b) and (c) are due to Chen, Gross and Rieper [7]: Theorem 6.5 Let G be a graph.

(a) If H is a subgraph of G, then γavg(H ) ≤ γavg(G). ( ) G γ (G) ≥ 1 γ (G) b If is 3-regular, then avg 2 max . ( ) G γ (G) ≥ 1 β(G) β(G) c If is a simple, 2-connected graph, then avg 16 , where is the cycle rank.

7. Stratiﬁcation of embeddings

Superimposing an adjacency structure on the distribution of orientable embeddings was first explored as a possible aid to calculating the minimum genus and maximum genus of a graph. It subsequently appeared to offer some insight into the problem of deciding whether two graphs are isomorphic. For any graph G, the stratified graph SG is defined as follows. The vertices of SG are the orientable embeddings of G. Two embeddings ι1 and ι2 are V -adjacent if there is a vertex v of G for which moving a single edge-end at v is sufficient to transform a rotation system for ι1 into a rotation system for ι2; they are E-adjacent if there is an edge e of G for which moving both ends of e can transform a rotation system for ι1 into a rotation system for ι2. In either case, the two embeddings are adjacent. The induced subgraph of SG on the set of embeddings into the surface Sh is called the hth stratum of SG and is denoted by ShG. The cardinality of ShG is clearly gh(G). Gross and Tucker [16] proved that trying to descend to the minimum genus can be obstructed by traps, which are false minima, from which it is impossible to descend to a global minimum without first ascending. The depth of a trap is the minimum increment in genus that one must ascend on any path to the true minimum. 60 Jonathan L. Gross

Traps of arbitrary depth were subsequently constructed by Gross and Rieper [13]. This is consistent with Thomassen’s result [32] that the minimum genus problem is NP-complete. However, Gross and Rieper [13] also proved that no false maxima exist, so that it is possible to ascend from any embedding to a maximum embedding, even though strict ascent might not always be possible. This is consistent with the work of Furst, Gross and McGeoch [9] that establishes a polynomial-time algorithm for the maximum genus. Gross and Tucker [18] proved that the link of each vertex of the stratiﬁed graph is a complete invariant of the graph. They also demonstrated how two graphs with similar genus distributions may have markedly different embedding strata. These ﬁndings support the plausibility of a probabilistic approach to graph isomorphism testing, based on the sampling of higher-order embedding distribution data.

References

1. J. Chen, A linear-time algorithm for isomorphism of graphs of bounded average genus, SIAM J. Discrete Math. 7 (1994), 614–631. 2. J. Chen and J. L. Gross, Limit points for average genus (I): 3-connected and 2-connected simplicial graphs, J. Combin. Theory (B) 55 (1992), 83–103. 3. J. Chen and J. L. Gross, Limit points for average genus (II): 2-connected non-simplicial graphs, J. Combin. Theory (B) 56 (1992), 108–129. 4. J. Chen and J. L. Gross, Kuratowski-type theorems for average genus, J. Combin. Theory (B) 57 (1993), 100–121. 5. J. Chen and J. L. Gross, No lower limit points for average genus, Graph Theory, Combinatorics, and Algorithms, Vol. 1, Wiley (1995), 183–194. 6. J. Chen, J. L. Gross and R. G.Rieper, Overlap matrices and total imbedding distributions, Discrete Math. 128 (1994), 73–94. 7. J. Chen, J. L. Gross and R. G. Rieper, Lower bounds for the average genus, J. Graph Theory 19 (1995), 281–296. 8. R. A. Duke, The genus, regional number, and Betti number of a graph, Canad. J. Math. 18 (1966), 817–822. 9. M. L. Furst, J. L. Gross and L.A. McGeoch, Finding a maximum genus graph imbedding, J. Assoc. Comput. Mach. 35 (1988), 523–534. 10. M. L. Furst, J. L. Gross and R. Statman, Genus distribution for two classes of graphs, J. Combin. Theory (B) 46 (1989), 22–36. 11. J. L. Gross and M. L. Furst, Hierarchy for imbedding-distribution invariants of a graph, J. Graph Theory 11 (1987), 205–220. 12. J. L. Gross, E. W. Klein and R. G. Rieper, On the average genus of a graph, Graphs and Combinatorics 9 (1993), 153–162. 13. J. L. Gross and R. G. Rieper, Local extrema in genus-stratiﬁed graphs, J. Graph Theory 15 (1991), 159–171. 14. J. L. Gross, D. P. Robbins and T. W. Tucker, Genus distributions for bouquets of circles, J. Combin. Theory (B) 47 (1989), 292–306. 15. J. L. Gross and T. W. Tucker, Generating all graph coverings by permutation voltage assignments, Discrete Math. 18 (1977), 273–283. 3 Distribution of embeddings 61

16. J. L. Gross and T. W. Tucker, Local maxima in graded graphs of imbeddings, Ann. New York Acad. Sci. 319 (1979), 254–257. 17. J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, 1987, and Dover, 2001. 18. J. L. Gross and T. W. Tucker, Stratiﬁed graphs for imbedding systems, Discrete Math. 143 (1995), 71–86. 19. D. M. Jackson, Counting cycles in permutations by group characters, with an application to a topological problem, Trans. Amer. Math. Soc. 299 (1987), 785–801. 20. J. Keilson and H. Gerber, Some results for discrete unimodality, J. Amer. Statist. Assoc. 66 (1971), 386–389. 21. J. H. Kwak and J. Lee, Enumeration of graph embeddings, Discrete Math. 135 (1994), 129–151. 22. B. Mohar, An obstruction to embedding graphs in surfaces. Discrete Math. 78 (1989), 135–142. 23. L. A. McGeoch, Algorithms for two graph problems: computing maximum-genus imbedding and the two-server problem, Ph.D. thesis, Carnegie-Mellon University, 1987. 24. B. G. Mull, R. G. Rieper and A. T. White, Enumerating 2-cell imbeddings of complete graphs, Proc. Amer. Math. Soc. 103 (1988), 321–330. 25. E. A. Nordhaus, R. D. Ringeisen, B. M. Stewart and A. T. White, On the maximum genus of a graph, J. Combin. Theory (B) 12 (1972), 260–267. 26. R. G. Rieper, The enumeration of graph embeddings, Ph.D. thesis, Western Michigan University, 1990. 27. G. Ringel, Map Color Theorem, Springer-Verlag, 1974. 28. S. Stahl, Region distributions of graph embeddings and Stirling numbers, Discrete Math. 82 (1990), 57–78. 29. S. Stahl, Permutation-partition pairs III: Embedding distributions of linear families of graphs, J. Combin. Theory (B) 52 (1991), 191–218. 30. S. Stahl, Region distributions of some small diameter graphs, Discrete Math. 89 (1991), 281–299. 31. E. H. Tesar, Genus distribution of Ringel ladders, Discrete Math. 216 (2000), 235–252. 32. C. Thomassen, The graph genus problem is NP-complete, J. Algorithms 10 (1989), 568–576. 33. N. H. Xuong, How to determine the maximum genus of a graph, J. Combin. Theory (B) 26 (1979), 217–225. 4 Algorithms and obstructions for embeddings

BOJAN MOHAR

1. Introduction 2. Planarity 3. Outerplanarity and face covers 4. Disc embeddings and the 2-path problem 5. Graph minors and obstructions 6. The algorithms for embeddability in general surfaces 7. Computing the genus References

This chapter gives a brief introduction to algorithms for ﬁnding embeddings of graphs in surfaces and discusses obstructions that prevent a graph from having an embedding. It starts with questions related to planarity and continues with obstructions to outerplanar graphs and the 2-path problem. Algorithms and obstructions for embeddings of graphs in surfaces of higher genus are more complicated, and known results are surveyed.

1. Introduction

In this chapter we study the theory of embeddings of graphs in surfaces, algorithms for ﬁnding embeddings, and the related questions about obstructions for the existence of embeddings. These are interesting from the theoretical viewpoint in mathematics and in theoretical computer science and appear in the study of algorithms and computational complexity. But these results are important also in the treatment of some practical problems. Planar graphs and their drawings in the plane occur naturally in a variety of applications. With the theoretical developments of computer science, graph planarity has received central attention in several areas of research. The following are some of the ﬁelds in which planar graphs often appear:

62 4 Algorithms and obstructions for embeddings 63

• VLSI design: Planar layouts are of great importance in the design of large-scale integrated circuits, since planar layouts of electrical networks are needed for the realization of complex electronic circuits. This has led to the development of algorithms for producing such layouts, the study of crossing numbers of graphs [13], and routing and linking problems (see, for example, [28] and [42]). • Theory of algorithms: Many computational problems on graphs have nice solutions when restricted to planar graphs. In such algorithms, the planar structure can be effectively used. Such problems include the partitioning of graphs (related to the celebrated separator theorem of Lipton and Tarjan [45]), with applications to divide-and-conquer algorithms, the graph isomorphism problem, subgraph isomorphism, shortest path problems and edge-colouring. Several basic problems with speciﬁc solutions on planar graphs are presented in the monograph by Nishizeki and Chiba [55]. • Graph drawing: Visual representation of data frequently requires drawings on a computer screen where, for obvious reasons, planar drawings are desirable. There is a vast literature on graph drawing: see Chapter 8 and the monograph by Di Battista et al. [25].

All of these developments point to the importance of efficient algorithms for planarity testing and for constructing plane embeddings of graphs. The theoretical counterpart to finding actual embeddings is the study of obstructions, the simplest substructures whose presence guarantees that a given graph cannot be embedded as desired. Embeddings of graphs in surfaces more general than the plane were originally influenced by more theoretical problems. Their early motivation is undoubtedly the Heawood problem (see Gross and Tucker [33] or Mohar and Thomassen [54]) on colourings of graphs embeddable in a fixed surface.Additional motivation came from Robertson and Seymour’s theory of graph minors (see Kawarabayashi and Mohar [39]). Similarly, as for planar graphs, the graphs of bounded genus form a big class of graphs on which many computational problems can be efficiently solved. Some problems that have been studied for graphs of bounded genus are graph partitioning (with application to the divide-and-conquer approach), edge-separators, crossing number problems, maximum independent sets, graph isomorphism, edge-colouring, max-cut and the enumeration of matchings, shortest path problems, dominating set, etc. For the time complexity of graph embedding algorithms, nearly all authors assume the random-access machine (RAM) model, with unit cost for certain basic operations. This model of computation was introduced by Cook and Reckhow [22], and is known as the unit-cost RAM. In this model, operations on integers, whose value is O(I)(where I is the size of the input), need only constant time. (We 64 Bojan Mohar note that the time complexity in this model may differ from that in the standard random-access machine model [1] by a factor of log I.) It is instructive to observe the following result that easily follows from Euler’s formula. Recall that the Euler genus of the surface S is defined as γ (S) = 2−ε(S), where ε(S) is the Euler characteristic of S.IfG is a graph with n vertices and m edges, embedded in a surface of Euler genus h, then

m ≤ 3n + 3(h − 2).

By this result, graphs of order n and genus h contain O(n+h) edges. In particular, if h is constant then m is proportional to n, and hence it is customary to express the time complexity in terms of n instead of the input size.

2. Planarity

In 1930, Kuratowski [43] published a beautiful result that characterizes obstructions to planarity. Let us recall from the Introduction that Kuratowski’s theorem states that a graph cannot be embedded in the plane if and only if it contains a subgraph that is isomorphic to a subdivision of K5 or K3,3. In other words, K5 and K3,3 are the only (minimal) obstructions to planarity. There are many equivalent formulations of this celebrated theorem that show the central role played by Kuratowski’s theorem in the theory of planar graphs. The reader may consult [54] for an extensive treatment of this topic. It was discovered rather early that the planarity of graphs can be checked in polynomial time: Auslander and Parter [12] presented an O(n3) algorithm, which was improved to O(n2) in [24], [32]. A graph is planar if and only if each of its blocks is. Furthermore, the blocks of a graph can be found in linear time by a simple modification of a depth-first search. What is less obvious is that the same statements also hold for the 3-connected components of a graph. Hopcroft and Tarjan [35] found a complicated linear-time procedure for obtaining the 3-connected components of a graph in linear time. This gave rise to the first linear-time algorithm for testing planarity [36]. Another important planarity testing algorithm was found by Lempel, Even and Cederbaum [44], who introduced a useful tool, which we now describe. Given a pair of adjacent vertices s and t in a graph G, an ordering v1,v2,...,vn of the vertices of G is called an st-ordering if v1 = s, vn = t, and for every i = 2, 3,...,n− 1, vi has neighbours vj and vl with j

Lemma 2.1 A graph with at least three vertices admits an st-ordering if and only if it is 2-connected. Furthermore, if G is 2-connected, there is an st-ordering for every pair of adjacent vertices s and t.

The main idea of Lempel, Even and Cederbaum was that, for any embedding of the subgraph Gi of G induced by v1,v2,...,vi (1 ≤ i

Theorem 2.2 There is a linear-time algorithm which, for any given graph G, either returns an embedding in the plane by specifying a corresponding rotation system, or returns a subgraph of G that is homeomorphic to K5 or K3,3.

One of the ﬁrst planarity algorithms was described by Tutte and appeared in his seminal paper ‘How to draw a graph’ [64]. It was described only for 3- connected graphs, but can be upgraded to arbitrary planar graphs, by passing ﬁrst to the 2-connected components and then to the 3-connected components of the graph. So, suppose that G is a 3-connected graph and that C = v1v2 ...vkv1 is an induced non-separating cycle of G.IfG is planar, then C is necessarily a facial cycle, by the Jordan curve theorem. Let A1,A2,...,Ak be the consecutive vertices of a convex k-gon in the plane. The main idea of Tutte was to look for a layout of vertices of G in the plane such that

(i) each vertex vi is mapped to Ai; (ii) each vertex not on C is placed at the barycentre of the positions of its neighbours. Considering the coordinates of vertices not on C as unknowns, we can express (ii) as a system of 2(n−k)linear equations in 2(n−k)unknowns. Tutte proved that this system has a unique solution, and that the solution provides a planar straight-line convex embedding of G if and only if G is planar. 66 Bojan Mohar

Various reincarnations of Tutte’s procedure are known as spring embedding algorithms, and are widely used to produce nice drawings of planar graphs; see, for example, Chapter 8 or the monograph by Di Battista et al. [25]. Speaking of ‘nice’ straight-line embeddings of planar graphs, we should mention circle packing representations, whose use enabled Brightwell and Scheinerman [20] to prove the following: Theorem 2.3 Let G be a 3-connected planar graph. Then there exist simultaneous straight-line drawings of G and its geometric dual G∗ in the plane (where the vertex of G∗ representing the unbounded face lies at inﬁnity) for which only dual edges cross, and such edges intersect at right angles. It was proved by Mohar [49] that an arbitrarily good approximation to a ‘primal- dual’ representation from Theorem 2.3 can be obtained in polynomial time. It can be shown that exact coordinates are not always rational, so approximations are unavoidable. It is worth mentioning that this algorithm extends to 3-connected graphs on arbitrary surfaces [49].

3. Outerplanarity and face covers

Agraph, of order greater is outerplanar if it can be embedded in the plane such that all vertices lie on the unbounded face. Let G+ be the graph obtained from G by adding a new vertex that is adjacent to all vertices of G. Clearly, G is outerplanar if and only if G+ is planar. This observation yields one possible linear-time algorithm for testing whether a graph is outerplanar (by using a planarity testing algorithm). However, there are simpler direct algorithms (see [19]). Interestingly, the two graphs obtained from the Kuratowski graphs by deleting one vertex (see Fig. 1) are the only obstructions to outerplanarity. This follows easily from Kuratowski’s theorem. Theorem 3.1 A graph of order greater than 4 is outerplanar if and only if it contains no subgraph homeomorphic to K4 or K2,3.

K4 K2,3

Fig. 1. Obstructions for outerplanarity

Bienstock and Dean [14] studied the more general classes Out(k) of graphs that have an embedding in the plane, for which there exists a set of at most k faces whose 4 Algorithms and obstructions for embeddings 67 boundaries contain all vertices of the graph; observe that Out(1) is precisely the set of all outerplanar graphs. For each fixed k, the family Out(k) is minor-closed so the graphs in this class can be characterized by a finite set of excluded minors; this follows from the Robertson–Seymour theory of graph minors (see Chapter 5). In particular, the graphs in Out(k) are recognizable in polynomial time (see Theorem 6.1). Bienstock and Dean also obtained estimates of how large these excluded minors might be. Archdeacon et al. [5] determined all 38 excluded minors for the class Out(2). The class of graphs Out(k) is also related to the problem of which infinite planar graphs admit an embedding on the sphere with at most k accumulation points. Since the sphere is compact, but infinite locally finite graphs are not, such graphs must have at least one accumulation point. Halin [34] found the structure of minimal graphs for which every embedding on the sphere has at least two accumulation points.

Theorem 3.2 Let G be a connected locally finite infinite planar graph. Then G has an embedding in the sphere with one accumulation point if and only if G contains no subdivision of any of the four graphs in Fig. 2, where the arrows indicate one-way infinite paths in the graph.

Fig. 2. Halin’s obstructions for a single accumulation point

Extensions to more accumulation points and to similar problems on other surfaces have been considered in [6], [7] and [17]. It is possible to generalize the notion of outerplanar graphs to other surfaces. Given a surface S, a graph G is S-outer-embeddable in S if G can be embedded in S in such a way that some facial walk contains all vertices. It was shown by Archdeacon et al. [8] that there are 32 excluded minors for the class of projective planar outer-embeddable graphs. Archdeacon and Sagols [10] considered a similar class of planar graphs for which every embedding on the sphere has a vertex (or a face) that is not surrounded by k disjoint ‘nested’ cycles of the graph. They found the excluded minors for the class of plane graphs that have a face F for which all other faces have a vertex in common with F ; further details can be found in [3]. 68 Bojan Mohar

4. Disc embeddings and the 2-path problem

Another generalization of outerplanarity comes from the following problem. Let G be a graph and C be a cycle of G. Is there an embedding of G in a disc D such that C forms the boundary of D? Such an embedding is called a C-disc embedding of G. The problem of C-disc embeddability can be reduced to planarity testing of the graph obtained from G by adding a new vertex adjacent to all vertices on C.A related linear-time algorithm, which also discovers a more special obstruction if there is no C-disc embedding, was found by Mohar [47]. If G is a graph and C is a cycle of G, two paths P1,P2 in G are disjoint crossing paths with respect to C if they are disjoint, both P1 and P2 intersect C precisely at their end-vertices, and their end-vertices interlace on C (see Fig. 3(a)). A tripod on C is a subgraph T of G that consists of ﬁve vertices v1,v2,u1,u2,u3 and six internally disjoint paths Pij that join vi and uj , together with three vertex-disjoint paths Q1,Q2,Q3 joining u1,u2 and u3 with C and that intersect the union of the paths Pij only at u1,u2 or u3. Moreover, T intersects C only at the end-vertices of Q1,Q2 and Q3. One or more of the paths Qj is allowed to be a single vertex, in which case uj ∈ V(C)(see Fig. 3(b)).

v1 v2 P1 a c

u3 P u 2 Q1 1 u2 Q Q2 3

(a) (b)

Fig. 3. Disjoint crossing paths and a tripod

+ Theorem 4.1 Let G be a graph and C be a cycle of G. Let GC be the graph obtained from G by adding a new vertex v0 adjacent precisely to all vertices on C. Then G admits no C-disc embedding if and only if G contains a subgraph of one of the following three kinds:

• a pair of disjoint crossing paths with respect to C; • a tripod on C; 4 Algorithms and obstructions for embeddings 69

+ • a Kuratowski subgraph contained in a 3-connected component of GC different from the 3-connected component containing C and v0. Furthermore, there is a linear-time algorithm that either ﬁnds a C-disc embedding or returns a subgraph described above.

This theorem is closely related to the 2-path problem: given four vertices s1,s2,t1,t2 in a graph G, do there exist disjoint s1-t1 and s2-t2 paths? This problem was essentially solved by Jung [37], who proved the following beautiful result.

Theorem 4.2 Let s1,s2,t1 and t2 be four vertices in a 4-connected graph G. Then there exist disjoint s1-t1 and s2-t2 paths unless G is planar and all four vertices lie on the boundary of some face in the interlaced order s1,s2,t1,t2.

A k-separation of G is a pair (G1,G2) of edge-disjoint subgraphs of G, each with at least k + 1 vertices, such that G = G1 ∪ G2 and |V(G1 ∩ G2)|=k. This k-separation is proper if, for any two vertices v and w in G1 ∩ G2, there is a vw-path in G2 with no intermediate vertex in G1 ∩ G2. Let G be a 2-connected graph and let s1,t1,s2 and t2 be four vertices of G.IfG has a proper k-separation (G1,G2) such that k ≤ 3 and all four vertices belong to G1, then replace G by the graph obtained from G1 by forming a complete graph on G1 ∩ G2. Repeat this procedure as long as such k-separations exist. The resulting graph is called the torso of G (with respect to s1,t1,s2,t2).

Theorem 4.3 Let G be a 2-connected graph, let s1,t1,s2,t2 be distinct vertices of G, and let G be the corresponding torso. Then G does not contain disjoint s1-t1 and s2-t2 paths if and only if G is planar and the vertices s1,t1,s2 and t2 lie on the boundary of the same face in the interlaced order s1,s2,t1,t2.

Theorem 4.3 gives rise to a quadratic-time algorithm (see Shiloach [59]) for the 2-path problem; it is not known whether the 2-path problem can be solved in linear time.

5. Graph minors and obstructions

Let S be a surface. A graph G without vertices of degree 2 is a minimal forbidden subgraph for S if G cannot be embedded in S but every proper subgraph of G can be embedded in S. The set of all minimal forbidden subgraphs for S is denoted by Forb(S). The graph G is a minimal forbidden minor (or an excluded minor) for S if G cannot be embedded in S, but every proper minor of G can be embedded in S. Let us denote the set of all excluded minors for S by Forb0(S). Clearly, Forb0(S) ⊆ Forb(S). 70 Bojan Mohar

A graph cannot be embedded in S if and only if it contains a subgraph that is a subdivision of a minimal forbidden subgraph for S. Similarly, a graph cannot be embedded in S if and only if it contains an excluded minor for S. The following proposition shows that Forb(S) is ﬁnite if and only if Forb0(S) is ﬁnite.

Theorem 5.1 For every graph H0, there is a ﬁnite collection H1,H2,...,Hp of graphs such that an arbitrary graph G has H0 as a minor if and only if G contains a subdivision of one of H1,H2,...,Hp.

The Kuratowski theorem states that for the plane, Forb0(S) = Forb(S) = {K5,K3,3}. Concerning other surfaces, a complete analogue of the Kuratowski theorem is known only for the projective plane N1. Glover, Huneke and Wang [31] presented a list of 103 minimal forbidden subgraphs, and Archdeacon [2] proved that this list is complete.

Theorem 5.2 A graph can be embedded in the projective plane if and only if it has no subgraph homeomorphic to one of 103 graphs.

Out of the 103 minimal forbidden subgraphs for the projective plane, the following 35 graphs are the excluded minors for the projective plane:

• the disjoint union of two Kuratowski graphs, 2K5,2K3,3 and K5 ∪ K3,3; • the one-vertex identiﬁcation of two Kuratowski graphs (there are three such graphs); • the six graphs in Fig. 4, obtained by identifying two vertices of two Kuratowski graphs and possibly deleting an edge; • the graphs in Fig. 5.

Fig. 4. Two-vertex identiﬁcations of Kuratowski graphs

Acomplete set of excluded minors is not known for any other surface. However, as a side result in their theory of graph minors, Robertson and Seymour [56] proved that there is an analogue of Kuratowski’s theorem for every surface. A short proof of this seminal result appeared in [54, Chap. 7] and can also be found in Chapter 5.

Theorem 5.3 For every surface S, the set of excluded minors is ﬁnite. 4 Algorithms and obstructions for embeddings 71

Fig. 5. The 3-connected excluded minors for the projective plane

In other words, for each surface S there exists a finite set H of graphs such that a graph can be embedded in S if and only if it contains no subgraph homeomorphic to one of the graphs in H. The proof of Theorem 5.3 published by Robertson and Seymour is not constructive, in the sense that it does not give a bound on the number of excluded minors. A constructive proof for non-orientable surfaces was published by Archdeacon and Huneke [9]. For the more complicated case of orientable surfaces, the first constructive proof was published by Mohar [50]. Little is known about the set of excluded minors for surfaces other than the sphere and the projective plane, although much work has been done for the torus S1. A subgraph H of G is a K2,3-subgraph in G if H consists of two vertices v,w and three internally disjoint paths P1,P2,P3 joining v and w, and some component of G − H is adjacent to interior vertices of all three paths. Similarly, H is a K4- subgraph in G if it is a subdivision of K4 and some component of G − H is adjacent to all four vertices of degree 3 in H. A subgraph of G is a K-graph if it is either a K2,3-orK4-subgraph. Decker [23] has classified 259 graphs in Forb(S1) that contain a subgraph that is the disjoint union of a subdivision of K5 or K3,3 and a K-graph. It is also known that precisely 270 minimal forbidden minors for the torus can be embedded in the projective plane. These are the graphs that can be obtained 72 Bojan Mohar from the projective 4 × 4 grid by Y-exchanges (see Fig. 6(a)). This family is disjoint from the family determined by Decker. It has been proved that Forb(S1) contains three graphs of order 8 and 48 of order 9. Using a computer, W. Myrwold has shown that Forb(S1) contains precisely 660 graphs with 10 vertices. Further computer search revealed a vast number of additional obstructions and, in particular, showed that |Forb(S1)| > 239 000 and |Forb0(S1)| > 16 600. Minimal forbidden graphs for the torus have also been studied by Bodendiek and Wagner [16]. They have reduced the number of necessary excluded minors by adding two further reductions, the Y and the bow-tie operations shown in Fig. 6. They expressed the opinion that the list of torus obstructions in [16] should be (almost) complete.

(a) (b)

Fig. 6. The Y and the bow-tie reductions

The constructive proofs of Theorem 5.3 give an upper bound on the size of graphs in Forb(S1). However, these bounds are enormous. Mohar and Thomassen [54, Problem 6.6.1] asked for a more realistic bound: does every minimal forbidden subgraph for the torus have fewer than 100 edges? Glover asked whether every edge of a minimal forbidden subgraph (for any surface S) is contained in a homeomorph of K5 or K3,3 (see [33, p. 53]). Brunet, Richter and Širánˇ [21] proved that this is so when S is non-orientable. For orientable surfaces, Širánˇ [60] proved that the same holds for 3-connected graphs. However, the minimal forbidden subgraph H for the torus shown in Fig. 7 contains the edge e which is not part of a K5 or K3,3 homeomorph in H [21]. This answers Glover’s question in the negative.

Fig. 7. A minimal forbidden subgraph for the torus 4 Algorithms and obstructions for embeddings 73

A more speciﬁc conjecture of Glover is still open: every minimal forbidden subgraph for the non-orientable surface Nk is the union of k + 1 subgraphs, each of which is homeomorphic to K5 or K3,3. For pseudosurfaces the situation changes, since the class of graphs embeddable in a given pseudosurface need not be closed under the taking of minors. Obstructions for pseudosurfaces were considered in [4], [15] and [61]. Širánˇ and Gvozdjak [61] considered the spindle surface, which is obtained from two spheres by identifying two pairs of points.

Theorem 5.4 The family of minimal forbidden subgraphs for the spindle surface contains inﬁnitely many 4-regular graphs.

The spindle surface is the simplest pseudosurface for which a Kuratowski theorem does not hold. Knor [41] proved the following result.

Theorem 5.5 If S is a pseudosurface that is not a surface and cannot be obtained by a tree-like one-point identiﬁcation process from spheres and at most one pseudosurface with one singular point, then S has inﬁnitely many minimal forbidden subgraphs.

6. Algorithms for embeddability in general surfaces

Determining the genus or the crosscap number of a graph is NP-hard, as we will discuss in Section 7. However, for a ﬁxed surface there is a polynomial-time algorithm for determining whether a given graph can be embedded in the surface; such algorithms were found ﬁrst by Filotti, Miller and Reif [30]. For the orientable αh+β surface Sh, their algorithm runs in time O(n ), where α and β are constants; in particular, for the torus the running time is O(n188). Later, Djidjev and Reif [27] announced an improvement by presenting an algorithm for embeddability in Sh, whose time complexity is 2(αh)!nβ , where α and β are constants. Robertson and Seymour also proved the following seminal result [57].

Theorem 6.1 For every graph M there exists a polynomial-time algorithm that for a given graph G of order n decides whether G contains M as a minor.

Robertson and Seymour originally presented an O(n3) algorithm and later improved it to O(n2 log n). Kawarabayashi and Reed recently announced a further improvement to O(nlog n). By Theorem 5.3, every surface has ﬁnitely many excluded minors. Theorem 6.1 therefore yields a polynomial-time algorithm for testing embeddability in a given surface. Similar algorithms (using a different set of excluded minors) can be used to recognize members in any minor-closed family of graphs in polynomial time. 74 Bojan Mohar

It is not hard to extend an algorithm for testing embeddability to construct an embedding if one exists. The idea is incrementally to join two edges incident with the same vertex and thus test whether there is an embedding where those two edges are consecutive in the local rotation around the vertex. This is repeated (polynomially many times) until a rotation system for the whole graph is obtained. A disadvantage of algorithms based on Theorems 5.3 and 6.1 is that they use the lists of excluded minors that are not known for surfaces different from the 2-sphere and the projective plane. Even for the projective plane, whose excluded minors are known (Theorem 5.2), algorithms based on checking for the presence of excluded minors are rather time consuming, since their running-time estimates involve enormous constants. Linear-time algorithms have been devised for embedding graphs in the projective plane (Mohar [46]) and in the torus (Juvan et al. [38]). Finally, Mohar [48], [50] proved that for every ﬁxed surface S, embeddability in S can be decided in linear time.

Theorem 6.2 For each surface S, there is a linear-time algorithm for testing embeddability in S. Moreover, for every input graph G, the algorithm either ﬁnds an embedding of G in S, or ﬁnds a subgraph homeomorphic to a member of Forb(S).

The basic technique used in [30] and [27] is to consider all possible embeddings of a forbidden subgraph F ⊆ G for a surface of smaller genus. An attempt is made to extend each such partial embedding of F in the surface S to an embedding of the whole graph. In [50], the same technique is used recursively to obtain an embedding in S, or to extend the subgraph so that its embeddings in S become more and more restricted. After a bounded number of steps, either an embedding is found or an obstruction is produced. By studying the structure of such obstructions, it is shown that they can always be changed into obstructions with only a bounded number of vertices of degree greater than 2. Aside result of the proofs in [50] (and some accompanying papers cited therein) is that the returned minimal forbidden subgraph is homeomorphic to a graph with a bounded number of edges, where the bound depends only on S. This yields a constructive proof of the result of Theorem 5.3; see the discussion in Section 5. The results and methods used to prove Theorem 6.2 can be used towards solving more general embedding extension problems. Here one has a ﬁxed embedding of a subgraph K of G in some surface and asks for embedding extensions to G or (minimal) obstructions for the existence of such extensions. For some embedding extension problems the number of minimal obstructions is inﬁnite. 4 Algorithms and obstructions for embeddings 75

7. Computing the genus

The problem of determining the genus of the graph is hard. Speciﬁcally, Thomassen [62] proved the following result:

Theorem 7.1 The following three problems are NP-complete:

• Does a given graph triangulate some surface? • Does a given graph triangulate some orientable surface? • Does a given graph triangulate some non-orientable surface? The similar problem of determining whether a graph admits an embedding of face-width at least 3 in some surface was also shown to be NP-complete [52]. The genus problem asks whether γ (G) ≤ k, where the graph G and the integer k are part of the input. If we replace the genus γ (G) by the non-orientable genus or by the Euler genus, we obtain the non-orientable genus problem and the Euler genus problem, respectively.

Corollary 7.2 The genus problem, the non-orientable genus problem and the Euler genus problem are all NP-complete.

Thomassen [63] proved that both the genus and the non-orientable genus problem remain NP-complete even if the input is restricted to cubic graphs. Several hard problems are polynomially solvable in some minor-closed families of graphs. It would be interesting to know more about which minor-closed families have this property. Suppose that F is a family of graphs such that a ﬁxed graph M is not a minor of any member of F. By the excluded minor theorem of Robertson and Seymour, graphs in F have a ‘tree-like’ structure of graphs that are ‘nearly embeddable’ in some surface in which H cannot be embedded (see [39] for more details on this deep result). This structure gives rise to three ‘generic’ minor-closed families of graphs:

• graphs of bounded tree-width; • graphs embeddable in a ﬁxed surface; • graphs that become planar after the deletion of a bounded number of vertices. For graphs of bounded tree-width, almost all known optimization problems are solvable in polynomial time. It is also easy to ﬁnd a polynomial-time algorithm for computing the non-orientable genus and the Euler genus. The case of the orientable genus is more demanding and has been solved only recently (see the discussion in [40]).

Theorem 7.3 The genus of graphs of bounded tree-width can be computed in linear time. 76 Bojan Mohar

On the other hand, the minor-closed families of type (3) are tougher. A graph G is an apex graph if it contains a vertex v for which G − v is planar. It is proved in [51] that the genus of apex graphs is hard to compute.

Theorem 7.4 The genus problem is NP-complete for apex graphs.

For graphs embeddable in a fixed surface, the genus problem may seem easy. Indeed, for graphs embeddable in a fixed orientable surface, the genus can be computed in linear time (by applying the algorithm of Theorem 6.2). However, computing the orientable genus of graphs embeddable in a fixed non-orientable surface is more of a mystery. It has been known since the 1960s that graphs embeddable in even the projective plane can have arbitrarily large genus (see Auslander, Brown and Youngs [11]). This phenomenon is now appropriately understood ever since Fiedler, Huneke, Richter and Robertson [29] proved that the genus of a projective planar graph depends only on the face-width of its embedding in the projective plane.

Theorem 7.5 Let r be the face-width of a graph G that is embedded in the projective plane. If r = 2, then

γ (G) = 1 r . 2 (1)

If r = 2, then γ (G) is either 0 or 1.

This result implies that the genus of graphs embeddable in the projective plane can be computed in quadratic time. Theorem 7.5 has been generalized to the Klein bottle by Robertson and Thomas [58], as follows. Let us consider an embedding of G in N2.Ifα is a closed curve in the surface, we let cr(α, G) denote the number of points where α intersects G. (G) 1 (α, G) Denote by ord2 the minimum of 2 cr taken over all non-contractible and non-separating 2-sided simple closed curves α. Similarly, let ord1(G) denote 1 (α ,G) + 1 (α ,G) α ,α the minimum of 2 cr 1 2 cr 2 taken over all pairs 1 2 of non- homotopic 1-sided simple closed curves. The latter minimum, restricted to all (G) non-crossing pairs of 1-sided simple closed curves, is denoted by ord1 . Let

h = { (G), (G)} h = { (G), (G)}. min ord1 ord2 and min ord1 ord2 (2)

Robertson and Thomas [58] proved that if h ≥ 4, then γ (G) = h = h. Equations (1) and (2) imply that the genus of graphs that are embeddable in the projective plane or the Klein bottle can be computed in polynomial time. It is likely that the genus problem for graphs with bounded non-orientable genus is solvable in polynomial time, as conjectured by Robertson and Thomas. Indeed, they conjectured that (1) and (2) can be generalized as follows. 4 Algorithms and obstructions for embeddings 77

Suppose that C ={α1,α2,...,αp} is a set of closed curves in the surface Nk. Then C is crossing-free if the following hold: • no curve in C crosses itself (but ‘self-touching’ is allowed); • no two curves cross each other. α ,α ,...,α If there exist simple closed curves 1 2 p with pairwise-disjoint images in Nk such that each αi is homotopic to αi and such that every 1-sided closed curve N α ,α ,...,α in k crosses at least one of the curves 1 2 p, then we say that the family C is a blockage and that it blocks 1-sided curves in the surface. Suppose that a graph G is embedded in Nk and that C ={α1,α2,...,αp} is a blockage. The order of the blockage C is deﬁned as p (C,G)= 1 (k − p + s) + (α , G), ord 2 2 ord i (3) i=1 where s is the number of 1-sided closed curves in C, and 1 (α ,G) α 2 cr i if i is 1-sided, ord(αi,G)= 1 (α ,G) α 2 cr i if i is 2-sided. 1 (k − p + s) We note that the term 2 2 in (3) is an integer and that it is equal to the genus of the (bordered) orientable surface obtained by cutting Nk along the curves in . It is easy to prove (see [58]) that for each graph G embedded in Nk and every blockage C in Nk, we have

γ (G) ≤ ord(C, G). (4)

Based on (1), (2) and (4), one would expect that if G is a graph embedded in Nk with sufﬁciently large face-width, then the genus of G is related to the minimum order of a blockage in Nk. Mohar and Schrijver [53] proved that this is true up to a constant error term, even without the assumption on large face-width.

Theorem 7.6 Let G be a graph embedded in Nk, and let r be the minimum order of a crossing-free blockage. Then

r − (64k)2 − k + 1 ≤ γ (G) ≤ r.

It is worth mentioning that the non-orientable genus of graphs embeddable in a ﬁxed orientable surface Sh can be determined in linear time (again, by Theorem 6.2) since the non-orientable genus is bounded in terms of h; details may be found in [54, Ch. 4]. The results described above speak in favour of the following far-reaching general conjecture: 78 Bojan Mohar

Conjecture 7.7 Let M be a minor-closed family of graphs. If P = NP, then the genus problem for graphs in M is NP-complete if and only if M contains all apex graphs.

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R. BRUCE RICHTER

1. Introduction 2. Graph decompositions 3. Linked decompositions 4. Graphs with bounded tree-width 5. Finding large grids 6. Embedding large grids References

In their Graph Minors Project, Robertson and Seymour proved that, in any infinite set of graphs, one is a minor of another.In particular,if S is a surface, the set of minor-minimal graphs that are not embeddable in S is finite. Two central results of the Graph Minors Project are: • if the graphs in the infinite set have bounded tree-width, then one is a minor of the other; • graphs with large tree-width have large grids as minors. We present the ‘simple’ proofs of these two facts, and adapt an argument of Thomassen that shows how to apply them to prove the finiteness of the set of minor-minimal non-S-embeddable graphs.

1. Introduction

This chapter is a self-contained introduction to graph minors. It contains a complete proof of the generalization of Kuratowski’s theorem to higher surfaces; more importantly, it is a major step in understanding the whole Graph Minors Project of Robertson and Seymour. The only background needed is some familiarity with connectivity issues (essentially variations of Menger’s theorem and a willingness to view cutsets from different perspectives). Our experience with the arguments

81 82 R. Bruce Richter presented here is that we need to be able to focus on both the big picture and on the details. There are many small points that require their own little arguments and we have attempted to provide these in sufficient detail to make it easier not to lose sight of the big picture. A reflexive, transitive relation ≤ on a set X is a well-quasi-order if there is no infinite descending chain x1 >x2 >x3 >... and there is no infinite antichain, an infinite set of pairwise incomparable elements. An equivalent formulation is that if x1,x2,... is an infinite sequence of elements of X, then there exist infinitely i

• T contains an infinite descending chain t1 >t2 >t3 >... ; • T contains an infinite antichain; • T contains an infinite ascending chain t1

In the context of

Theorem 1.1 There is no inﬁnite antichain for

Roughly speaking, here is how the proof of Theorem 1.1 goes. Let G be an inﬁnite set of graphs. If the graphs in G have bounded tree-width (a notion we deﬁne in the next section), then the result is a consequence of the following important step, which we prove in Section 2.

Theorem 1.2 For each positive integer k, the relation

Thus, we can assume the graphs in G do not have bounded tree-width. The k-grid is the Cartesian product Pk Pk. The second important fact is:

Theorem 1.3 For each positive integer k, there is an integer f(k)such that every graph with tree-width of at least f(k)has a k-grid minor. 5 Graph minors: generalizing Kuratowski’s theorem 83

This theorem has been given a relatively elementary proof by Diestel, Gorbunov, Jensen and Thomassen [3]; we present their proof in Section 5. It shows that if the graphs in G do not have bounded tree-width, then they have arbitrarily large grids. This provides a structure that we can latch on to as we try to find minors. Here is one example of how to use Theorem 1.3 in the proof of Theorem 1.1. Suppose that H is a fixed graph, that G is a graph that contains a k-grid as a minor and that, within G, there are r pairwise-disjoint paths whose ends are in the grid minor but are otherwise disjoint from the grid minor. Furthermore, we assume that no two of the 2r ends of the paths are ‘close’ in the grid. If k and r are sufficiently large (depending on H), then G contains H as a minor. Thus, in order for G ={G1,G2,...} to have no element Gi (i>1) containing G1 as a minor, the elements of G with very large grids cannot have such a set of paths. This imposes restrictions on how the rest of Gi is attached to the grid minor, and the additional structure allows us to look for even more structure (this goes on about four levels deep), until eventually we deduce that G has two elements, one a minor of the other. These remarks emphasize that Theorems 1.2 and 1.3 are fundamental building blocks of the Graph Minors Project. Thomassen [10] observed the following, which we prove in Section 6.

Theorem 1.4 Let S be a ﬁxed surface. Then there is an integer f(S)such that if G is minor-minimal for the property of not embedding in S, then G does not contain an f(S)-grid as a minor.

To see the relevance of this observation, let F(S) denote the set of all graphs G that minor-minimally do not embed in S. Note that no element of F(S) is a proper minor of another element of F(S). By Theorem 1.4, the elements of F(S) have bounded sized grid minors. By Theorem 1.3, the graphs in F(S) have bounded tree-width. Finally, by Theorem 1.2, there can only be ﬁnitely many elements in F(S). Let ir(S) denote the set of all graphs G, with minimum degree at least 3, such that G does not embed in S but every proper subgraph of G does embed in S. The distinction here is that the graphs minimally do not embed in S, up to subdivision, rather than being minor-minimal.

Corollary 1.5 For each surface S, ir(S) is ﬁnite.

The proof is quite simple. If G ∈ ir(S), then it does not embed in S and so has an element H of F(S) as a minor. To get from G to H we can do only contractions (since contractions and deletions commute), so that G is obtained from H by splitting vertices. But there are only ﬁnitely many graphs with minimum degree 3 that can be obtained by splitting vertices in a given graph. 84 R. Bruce Richter

2. Graph decompositions

The main point of the next three sections is to show that the set of graphs with bounded tree-width are well-quasi-ordered with respect to minors. These sections are based on [4], [2] and conversations with Jim Geelen. Two main decompositions are used in this context: tree decompositions and branch decompositions. Until now, these have been treated as different, but related. Tree decompositions are more natural in the context of graphs, but branch decompositions have been successful for simplification and extension to matroids. A simultaneous generalization of these two concepts shows that they are more closely related than previously thought. We present this common general decomposition and prove the basic facts required for well-quasi-ordering. This also clarifies the relationship between the two decompositions. We first introduce a decomposition of a graph G and describe the relation to branch and tree decompositions of G. Section 3 contains the proof that, given any decomposition of a graph G, there is another decomposition that is not ‘worse’ than the original decomposition with respect to any of three parameters, and such that the second decomposition satisfies the Menger property of being ‘linked’ (to be defined in that section). Linked decompositions are used in Section 4 to show that any set of graphs having bounded tree-width is well-quasi-ordered. A decomposition of a graph G is a pair (T , f ), where T is a tree with a set L of leaves (vertices of degree 1), and f : E(G) → L is an injection. It is easy to understand branch decompositions from this point of view: a branch decomposition of a graph G is a decomposition (T , f ) for which T has maximum degree at most 3. Tree decompositions are more complicated. A tree decomposition of a graph G is a pair (T , W), where T is a tree and W is a set of subsets Wt of V (G), one for each vertex t of T , such that: (a) t Wt = V (G); (b) for each edge e = vw of G, there is a t ∈ V(T)such that v,w ∈ Wt ; (c) for each vertex v of G, the subtree of T induced by {t: v ∈ Wt } is connected.

It is not obvious how tree decompositions relate to decompositions. Let (T , f ) f be a decomposition. For each non-leaf vertex t ∈ V(T), let Wt denote the set of vertices v incident with edges e and e of G for which f(e)and f(e) are in different components of T − t. Furthermore, for any leaf t of T , either t ∈ f (E(G)),in f −1 f which case Wt consists of the ends of f (t),ort/∈ f (E(G)) and Wt = ∅.

Lemma 2.1 Let G be a graph without isolated vertices and let (T , f ) be a f decomposition of G. Then (T , {Wt : t ∈ V(T)}) is a tree decomposition of G. 5 Graph minors: generalizing Kuratowski’s theorem 85

f Proof The deﬁnition of Wt implies (a) and (b). As for (c), let v be any vertex of f f G and suppose t and t are vertices of T for which v ∈ Wt ∩ Wt . Let t1 be any f vertex on the path in T between t and t . We show that v ∈ Wt . Note that we may 1 assume that t, t and t1 are all distinct. Let e be an edge of G incident with v for which f(e)is in a component of T −t not containing t, and let e be an edge of G incident with v such that f(e) is in a component of T − t not containing t. Clearly, f(e)and f(e) are in different components of T − t1, since t and t are in different components of T − t1. Hence v ∈ W f t1 , as required.

The restriction to graphs without isolated vertices in Lemma 2.1 is not really important. The main aim of this section is to prove that if G is an inﬁnite set of graphs, each with bounded tree-width, then there exist G, H ∈ G such that G

Lemma 2.2 Let (T , f ) be a decomposition of a graph G and let t ∈ V(T). Then

• if τ ∈ δT (t), (Eτ ) ⊆ Xt ; X (E ) • each vertex in t is in at least two of the τ ; ∗ • τ ∈ δ (t) X = ∗ (E ) if T , t τ∈δT (t)\{τ } τ ; • ζ(T,f) ≤ ρ(T,f) ≤ 1 (T )ζ(T , f ) 2 . For a graph G, ρ(G) is the minimum of ρ(T,f), taken over all decompositions (T , f ) of G. For a tree decomposition (T , W), w(T, W) is the maximum of |Wt |, taken over all vertices t of T . The tree-width tw(G) of a graph G is the minimum of w(T, W), taken over all tree decompositions (T , W) of G. (Note that this is 1 more than the standard definition; since we are concerned only with graphs with bounded tree- width, this difference is not relevant. The main point of using (max |Wt |) − 1in the standard definition seems to be so that trees have tree-width 1.) We can already see that the definition of a decomposition is simpler than that of a tree decomposition. In the main proofs to follow, it will therefore be helpful to work with decompositions, rather than tree decompositions. We note that the proof of Lemma 2.1 implies that tw(G) ≤ ρ(G). Our next result shows the reverse inequality.

Lemma 2.3 Let G be a graph with at least one edge and let (T , W) beatree decomposition of G. Then there is a decomposition (T ,f)such that ρ(T ,f)≤ w(T, W).

Proof For each edge e = vw of G, part (b) of the deﬁnition of tree decomposition implies that there is a vertex te of T such that {v,w}⊆Wt . For each vertex t of T , let Et denote the set of edges e of G such that te = t. We let T be the tree obtained from T by adding, for each vertex t of T , |Et | new neighbours of t, all leaves of T . We obtain the decomposition (T ,f)by setting, for each edge e of G, f(e)to be one of the new leaves adjacent to t,ife ∈ Et . Observe that if t is a vertex of T but not T , then ρ(t) ≤ 2. Now if t0 ∈ V(T), X ⊆ W ρ(t ) ≤|W | v ∈ X v we claim that t0 t0 and therefore 0 t0 . For if t0 , then is incident with edges e and e such that f(e)and f(e) are in different components of T − t0. Let t and t be the neighbours of f(e)and f(e) in T . By the deﬁnition of f , v ∈ Wt ∩ Wt . If either t or t is t0, then we are done. Since f(e) and f(e) lie in different components of T − t0, it follows that t and t lie in different components of T −t0, and therefore in different components of T − t0. Thus, t0 lies on the path in T between t and t . Since the subgraph of T induced by {s ∈ V(T): v ∈ Ws} is connected and v ∈ Wt ∩ Wt , it follows that v ∈ W t0 , as required. As an immediate consequence, we have the following important observation. 5 Graph minors: generalizing Kuratowski’s theorem 87

Theorem 2.4 For any graph G with at least one edge, tw(G) = ρ(G).

Thus, in order to prove that graphs with bounded tree-width are well-quasi-ordered, it sufﬁces to show that graphs with bounded ρ are well-quasi-ordered (and, for a given bound, we are dealing with exactly the same set of graphs). A fundamental (though very easy) property of decompositions is that they are ‘minor-monotone’. This is summarized in the following lemma, which we will not explicitly require here. Only in this statement and its proof do we use superscripts H and G to make it clear for which graph the relevant function is being applied.

Lemma 2.5 Let (T , f ) be a decomposition of a graph G, let H be a minor of G, and let L be the set of leaves of T . Then the decomposition (T , f ) of H with f : E(H) → L deﬁned by f (e) = f(e)is such that, for each τ ∈ E(T ) and each t ∈ V(T), ζ H (τ) ≤ ζ G(τ) and ρH (t) ≤ ρG(t).

Proof Any minor H of G is obtained by a sequence of the operations:

(a) delete an edge; (b) delete an isolated vertex; (c) contract an edge.

We shall show that the lemma holds in each case and the general result then follows by a trivial induction. (a) Let H = G − e . For an edge τ of T , let Eτ denote the set of edges displayed by τ in the decomposition (T , f ) of G, labelled so that e ∈ Eτ .Ifv ∈ H (Eτ \{e }), then v is incident with an edge in both Eτ \{e } and E(G)\Eτ , G H G so v ∈ (Eτ ). Thus, ζ (τ) ≤ ζ (τ). Similarly, if t ∈ V(T)and if v ∈ V(H)is incident with edges e1 and e2 of H such that f (e1) and f (e2) are in different components of T − t, then H G f(e1) and f(e2) are in different components of T − t. Thus, Xt ⊆ Xt ,so ρH (t) ≤ ρG(t). (b) Deleting isolated vertices has no impact on decompositions. (c) Let H be the graph obtained from G by contracting the edge e = uv, and let w be the vertex of H that replaces u and v.Ifτ ∈ E(T ), then H (τ) ⊆ (G(τ) ∪{w}) \{u, v}, and w ∈ H (τ) implies that at least one of u and v is in G(τ), whence ζ H (τ) ≤ ζ G(τ). Similarly, for t ∈ V(T), H G H Xt ⊆ (Xt ∪{w}) \{u, v}, and if w ∈ Xt , then at least one of u and v is in G H G Xt ,soρ (t) ≤ ρ (t). Let bw(G) denote the smallest value of ζ(T,f)over all branch decompositions of G. We conclude this section with some remarks relating bw(G) and tw(G). Suppose ﬁrst that (T , f ) is a decomposition of G with a vertex t of degree at least 4inT . Let S be any tree having only vertices of degrees 3 and 1, exactly degT (t) 88 R. Bruce Richter of which have degree 1. Let T be obtained from T by deleting t and identifying each neighbour of t in T with one of the leaves of S. Deﬁne f (e) to be f(e). If t ∈ V(T)\S, then ρT (t) = ρT (t). For s ∈ S,ifv is a vertex of G incident with edges e1 and e2 such that f (e1) and f (e2) are in different components of T − s, then f(e1) and f(e2) are in different components of T − v. Hence T T Xs ⊆ Xt ; that is, ρ(T ,f ) ≤ ρ(T,f), as claimed. This implies the following lemma, which we will use in the well-quasi-ordering argument.

Lemma 2.6 Every graph G has a decomposition (T , f ) for which (T ) ≤ 3 and ρ(T,f) = tw(G).

Let (T , f ) be a decomposition of G as in Lemma 2.6; then (T , f ) is also a branch decomposition of G. Since tw(G) = ρ(T,f) = max{|Xt |: t ∈ V(T)} and, for τ ∈ δ(t), (Eτ ) ⊆ Xt , we see that tw(G) ≥ bw(G). On the other hand, let (T , f ) be a branch decomposition such that bw(G) = ζ(T,f). By Theorem 2.4 and Lemma 2.2:

(G) = ρ(G) ≤ ρ(T,f) ≤ 3 ζ(T,f) = 3 (G) . tw 2 2 bw G (G) ≤ (G) ≤ 3 (G) Lemma 2.7 For any graph , bw tw 2 bw .

3. Linked decompositions

Robertson and Seymour originally proved that graphs with bounded tree-width are well-quasi-ordered, using a method that has not been published in its original form. In the meantime, Thomas [9] made a signiﬁcant improvement on a key idea and its proof, so the published version of [6] makes use of the improvement; this was further simpliﬁed by Diestel [2]. The purpose of this section is to prove the analogue of Thomas’s theorem for decompositions. The proof combines the techniques of [4] and [2]. In trying to prove that graphs with bounded tree-width are well-quasi-ordered, it is natural to try to ‘climb down’ the trees in tree decompositions of the graphs, showing that at each step along the way, the graphs we have built so far are well-quasi-ordered. As we look to go down to the next edge, we will be ‘gluing’ together all the graphs corresponding to the edges just above. The context will be simpler if we have only two edges above — this is Lemma 2.6 from the previous subsection. It is in the ‘climbing down’ process that we require some additional information about the decomposition, and it is the purpose of this section to provide this information. Let (T , f ) be a decomposition of a graph G. For distinct elements τ1 and τ2 of E(T ), let A(τ1,τ2) be the set of edges e of G such that f(e)is in the component 5 Graph minors: generalizing Kuratowski’s theorem 89

of T − τ1 not containing τ2, and let B(τ1,τ2) denote the set of edges e of G such that f(e) is in the component of T − τ2 not containing τ1. Let λ(τ1,τ2) be the maximum number of pairwise totally disjoint paths in G from (A(τ1,τ2)) to (B(τ1,τ2)). We abbreviate ‘pairwise totally disjoint’ to ‘disjoint’. On the other hand, let P(τ1,τ2) denote the minimal path in T containing both τ1 ∗ and τ2. Let λ (τ1,τ2) = min{ζ(τ): τ ∈ E(P(τ1,τ2))}. It follows from Menger’s ∗ theorem that λ (τ1,τ2) ≥ λ(τ1,τ2). The decomposition (T , f ) is linked if, for each pair τ1 and τ2 of edges of T , ∗ λ (τ1,τ2) = λ(τ1,τ2). We are now ready for the decomposition version of Thomas’s result. Theorem 3.1 Let (T , f ) be a decomposition of a graph G. Then there is a linked decomposition (T ,f) of G such that • (T ) ≤ (T ); • ζ(T,f) ≤ ζ(T,f); • ρ(T ,f) ≤ ρ(T,f).

Proof If (T , f ) is not already linked, then there exist τ1,τ2 ∈ E(T ) such that ∗ λ (τ1,τ2) > λ(τ1,τ2). Let A = A(τ1,τ2) and B = B(τ1,τ2). Note that λ(A) ≥ ∗ λ (τ1,τ2) > λ(τ1,τ2) ≥ 0, so A = ∅. Similarly, B = ∅. Two subsets E and E of E(G) cross if none of the four sets E ∩ E, E \ E, E \ E and E(G) \ (E ∪ E) is empty. Notice that if E crosses E, then each of E and E(G) \ E crosses each of E and E(G) \ E. Let X be a minimal set separating (A) and (B).Ifv is incident with an edge of A and w is incident with an edge of B, then any (v, w)-path P must contain a vertex of (A) and a vertex of (B). (The ﬁrst vertex of P incident with an edge not in A is in (A); the last vertex incident with an edge not in B is in (B).) Hence, it contains a vertex of X. Recall that, if W is a subset of V (G),abridge of W is a subgraph of G that is either an edge of G and its endpoints, both of which are in W, or a component K of G − W, together with all the edges from K to vertices in W, plus the endpoints of these edges. The preceding paragraph implies that no bridge of X in G contains both an edge of A and an edge of B. Let E denote the union of the edge-sets of the bridges of X that contain edges in A. Clearly, (E) separates (A) from (B), and (E) ⊆ X. The minimality of X implies that X = (E). Futhermore, A ⊆ E. Let E∗ be chosen so that: (i) A ⊆ E∗; (ii) (E∗) is a smallest set separating (A) from (B); ∗ (iii) subject to (i) and (ii), E crosses as few as possible of the sets Eτ displayed by edges of T . 90 R. Bruce Richter

There are three components in the graph T −{τ1,τ2}: the one containing f (A) is TA, the one containing f(B)is TB , and the third is TM (for ‘middle’). Thus, TA,M = (TA ∪ TM ) + τ1 is a component of T − τ2. Similarly, TM,B = (TM ∪ TB ) + τ2 is a component of T − τ1. We construct a decomposition (T ,f) of G as follows. Let T be constructed from disjoint copies TA,M and TM,B of TA,M and TB,M, respectively, with the copy ∗ of the end of τ2 in TA,M joined by a new edge τ to the copy of the end of τ1 in ∗ ∗ TM,B. The deﬁnition of f depends on whether e ∈ E or not. If e ∈ E , then ∗ f (e) is the copy of the vertex f(e)that is in TA,M, while if e/∈ E , then f (e) is the copy of the vertex f(e)that is in TM,B. We need to prove that (a) (T ) ≤ (T ); (b) ζ(T,f) ≤ ζ(T,f); (c) ρ(T ,f) ≤ ρ(T,f). In addition, we need to prove (d) (T ,f) is ‘better’ than (T , f ).

Observe that (a) is a triviality. For the rest, note that if τ ∈ E(TA) ∪ E(TB ) ∪ {τ1,τ2}, then it is trivial that ζ(τ ) = ζ(τ) for the copy τ of τ in T . A similar statement holds for vertices in V(TA) ∪ V(TB ). So let τ ∈ E(TM ). We conduct all of our arguments with the copy τA of τ in TA,M, realizing that the arguments for the copy τB of τ in TM,B follow by ∗ ∗ symmetry. Observe that Eτ = Eτ ∩E and Eτ is Eτ ∪E for τ ∈ E(P(τ1,τ2)), ∗ A B E Eτ \ E τ/∈ E(P(τ ,τ )) while τB is for 1 2 . Choose the labelling so that Eτ is the set of edges displayed by τ in (T , f ) such ∗ that Eτ ∩ B = ∅. Let P be a set of λ(E ) disjoint paths from (A) to (B). For ∗ each v ∈ (E ), there is a unique path Pv ∈ P containing v, and Pv consists of A B two subpaths – namely, Pv from the (A) end of Pv to v, and Pv from v to the (B) end of Pv. We begin with the following observation. ∗ ∗ Claim 1 The set (Eτ ) contains (Eτ ∩ E ) \ (E ) and, for each vertex v in ∗ ∗ B (Eτ ∩ E ) ∩ (E ), there is at least one vertex of Pv in (Eτ ). ∗ ∗ ∗ Proof If u ∈ (Eτ ∩E )\(E ), then u is incident with an edge e1 in Eτ ∩E and ∗ ∗ ∗ an edge e2 not in Eτ ∩ E .Ife2 ∈/ E , then u ∈ (E ), which is a contradiction. Hence e2 ∈/ Eτ ,sou ∈ (Eτ ). ∗ ∗ If v ∈ (Eτ ∩E )∩(E ), then v is incident with an edge in Eτ . Let b ∈ (B) B be the end of Pv ,sob is incident with an edge of B (which is certainly not in Eτ ). B Thus, Pv contains a vertex that is incident with an edge in Eτ and with an edge not in Eτ . 5 Graph minors: generalizing Kuratowski’s theorem 91

For each vertex t of TM , let τt ∈ δT (t) be the edge separating t from TB and T let δt = δT (t) \{τt }. By Lemma 2.2 the set Xt is the union, taken over τ ∈ δt ,of T the sets (Eτ ). For the copy tA of t in TA,M, we have the analogous set X . tA We have an immediate corollary to Claim 1. T T ∗ T ∗ Claim 2 For each t, Xt contains X \ (E ) and, for each v ∈ X ∩ (E ), tA tA B T there is at least one vertex of Pv in Xt . It follows immediately from Claims 1 and 2 that ζ(τA) ≤ ζ(τ) and ρ(tA) ≤ ρ(t), establishing (b) and (c). We now wish to prove (d), that (T ,f) is a ‘better’decomposition than (T , f ). For each decomposition (S, g) of a graph G, and for r ∈{0, 1, 2,..., |V (G)|}, let π(S,g,r) be the number of vertices s of S for which ρ(s) = r. We set

(S, g) = (π(S,g,|V (G)|), π(S,g,|V (G)|−1), ... , π(S,g,0)).

We shall prove that (T ,f) is lexicographically smaller than (T, f ). Here is the main point. T T T ∗ Claim 3 If t ∈ V(TM ) is such that |Xt |=|X | > 0, then X ⊆ (E ). tA tB What Claim 3 shows is that if r>|(E∗)|, then π(T ,f,r) ≤ π(T,f,r).We prove below that there is at least one value of r for which the inequality is strict. The main points in the proof of Claim 3 are the following three facts. Claim 4 Let A, B, C ⊆ E(G).IfA ∪ B crosses C, and if B crosses A but not C, then A crosses C. Proof We note that C \ A ⊃ C \ (A ∪ B) = ∅ and E(G) \ (A ∪ C) ⊃ E(G) \ ((A ∪ B) ∪ C) = ∅. It remains to show that (a) A ∩ C = ∅ and (b) A \ C = ∅. We note that

E(G) \ (B ∪ C) ⊇ E(G) \ ((A ∪ B) ∪ C) = ∅ and C \ B ⊇ C \ (A ∪ B) = ∅ . Since B does not cross C, either B ∩ C = ∅ or B \ C = ∅. Case 1. If B ∩ C = ∅, then

A ∩ C = (A ∪ B) ∩ C = ∅ , which is (a),

and A \ C ⊇ A ∩ B = ∅ , which is (b). 92 R. Bruce Richter

Case 2. If B \ C = ∅, then

A ∩ C ⊇ A ∩ B = ∅ , which is (a),

and A \ C = (A ∪ B) \ C = ∅ , which is (b). t T Et = E Et = E For each vertex of M , set τt ; note that τ∈δt τ . T T ∗ t ∗ Claim 5 If |Xt |=|X |, then |(E ∪ E )|=|(E )|. tA

∗ t ∗ Proof If u ∈ (E ∪ E ) \ (E ), then u is incident with some edge e1 not in ∗ t ∗ ∗ E ∪ E ; therefore, e1 is not in E . Since u/∈ (E ), u is not incident with any ∗ ∗ t t edge of E .Asu is also incident with an edge e2 ∈ E ∪E , it follows that e2 ∈ E , T T whence u ∈ (Eτ ), for some τ ∈ δt . By Claim 2 and the fact that |Xt |=|X |, tA B ∗ we see that u is a vertex of Pv for some v ∈ (E ) and that u is the only vertex B T ∗ t ∗ ∗ t from Pv in Xt . Thus, |(E ∪ E )|≤|(E )|. Since (E ∪ E ) separates (A) from (B), the choice of E∗ implies that |(E∗ ∪ Et )|≥|(E∗)|. T T ∗ ∗ t Claim 6 If |Xt |=|X | > 0 and Xt ⊆ (E ), then E crosses E . tA B Proof The set E(G) \ (E∗ ∪ Et ) contains B, so it is not empty. To see that E∗ ∩ Et = ∅, observe that T ∗ X = (Eτ ) = (E ∩ Eτ ). tA A τ∈δt τ∈δt

T ∗ Since X = ∅, there is a τ ∈ δt for which (E ∩ Eτ ) is not empty. For such a tA ∗ ∗ t τ, E ∩ Eτ = ∅, which in turn implies that E ∩ E = ∅. To see that E∗ \ Et = ∅, note that E∗ ⊆ Et implies that A ⊆ Et ⊆ E(G) \ B; thus, (Et ) separates (A) from (B). But E∗ ⊆ Et implies that (Et ) = (E∗ ∪ Et ) and, by Claim 5, (E∗ ∪ Et ) has the same size as (E∗). Thus, (Et ) is a smallest set separating (A) from (B) and is displayed in (T , f ). But our assumption is that no edge displayed by (T , f ) separating f (A) from f(B)has so small a label, so we have a contradiction. t ∗ T ∗ Finally, to see that E \ E = ∅, let w ∈ X \ (E ). There is some τ ∈ δt tB such that w is incident with an edge e1 ∈ Eτ and with an edge e2 ∈/ Eτ , where B B Eτ is the subset of E(G) \ B displayed by τB in (T ,f ). B ∗ ∗ E Eτ \ E Eτ ∪ E e e Since τB is either or , at least one of the edges 1 and 2 is not in E∗. But w/∈ (E∗),sow cannot be incident with any edges of E∗. Thus, ∗ t neither of the edges e1 and e2 is in E , and so e1 ∈ Eτ . Since Eτ ⊆ E , we have Et \ E∗ = ∅. 5 Graph minors: generalizing Kuratowski’s theorem 93

We are now ready for the proof of Claim 3. T ∗ ∗ Proof of Claim 3 If X is not a subset of (E ), then Claim 6 shows that E tB t t crosses E .Ifτ is any edge of T , then E and Eτ do not cross, because any two sets of edges displayed by (T , f ) are either disjoint or one is contained in the other. It ∗ t ∗ ∗ follows from Claim 4 that if E ∪ E crosses Eτ , then E crosses Eτ . Since E crosses Et but E∗ ∪ Et does not, it follows that (i) E∗ ∪ Et contains A; (ii) (E∗ ∪ Et ) is a smallest set separating (A) from (B); (iii) E∗ ∪ Et crosses fewer of the sets displayed by the edges of T than E∗ does. This contradicts the choice of E∗. We can now prove that (T ,f) is lexicographically smaller than (T, f ).We suppose not. For r>|(E∗)|, Claim 3 implies that π(T ,f,r) ≤ π(T,f,r), T ∗ T T since if |Xt |=r>|(E )|, then at most one of |X | and |X | can be r. tA tB Thus, our assumption that (T ,f) is not lexicographically smaller than (T, f ) T ∗ implies that if t ∈ V(TM ) is such that |Xt |=r>|(E )|, then exactly one of T T |X | and |X | is r. tA tB Let s and t be the ends of τ1 and τ2, respectively, in TM . Clearly (A) is a T T T T subset of both Xs and X ; similarly, (B) is a subset of both Xt and X . The sA tB choice of A and B implies that |(A)| > |(E∗)| and |(B)| > |(E∗)|, so both T T T B |X |=|Xt | and |X |=|Xt |. tA tB It follows that there is an edge τ = pq in TM ∩ P(τ1,τ2) such that:

• p is closer to s in TM than q is; T T • |X |=|Xp |; pA T T • |X |=|Xq |. qB T T ∗ By the choice of A and B, |Xp | and |Xq | are both greater than |(E )|. ∗ p p q p We show that E crosses E . We know that E = Eτ ⊆ E , A ⊆ E , ∗ ∗ T T E = Eτ ∩ E E = Eτ ∪ E (Eτ ) ⊆ X (Eτ ) ⊆ X τA , τB , p , and q . (a) Since A ⊆ E∗ ∩ Ep, E∗ ∩ Ep = ∅. (b) Since B ⊆ E(G) \ (E∗ ∪ Ep), E(G) \ (E∗ ∪ Ep) = ∅. ∗ p ∗ p T T (c) Suppose that E \ E = ∅. Then E ⊆ E . Because |X |=|Xp | > 0, pA T ∗ Claim 3 implies that X ⊆ (E ). Since pB

∗ p T ∗ (E ∪ E ) = (Eτ ) ⊆ X ⊆ (E ), B pB p ∗ p ∗ we conclude that (Eτ ) = (E ) = (E ∪ E ) ⊆ (E ). Since (Eτ ) is ∗ a set separating (A) from (B), it follows that |(Eτ )|≥|(E )|. Together 94 R. Bruce Richter

∗ with the previous containment, this implies that (Eτ ) = (E ). But (Eτ ) is displayed in (T , f ), contradicting the choice of A and B. (d) Finally, we show that Ep \ E∗ = ∅. For, otherwise, Ep ⊆ E∗. This is the p same argument, but applied to q. We have the relations (Eτ ) = (E ) = (Ep ∩ E∗) ⊆ XT ⊆ (E∗) E q (the last containment by Claim 3). Since τ A ∗ contains A and is disjoint from B, |(Eτ )|≥|(E )|, and again we have the ∗ ∗ p contradiction that (Eτ ) = (E ). Thus, E crosses E .

Note that p ∗ T ∗ (E ∪ E ) = (Eτ ) ⊆ X ⊆ (E ). B pB Therefore, (Ep ∪ E∗) = (E∗), and so (Ep ∪ E∗) is also a smallest set separating (A) from (B). The ﬁnal contradiction comes from the realization that E∗ ∪ Ep crosses fewer of the sets displayed by (T , f ) than E∗ does, since Claim 4 implies that everything crossed by E∗ ∪ Ep is also crossed by E∗; but E∗ crosses Ep, while E∗ ∪ Ep does not. To ﬁnish the proof of Theorem 3.1, choose a decomposition (T , f ) so that (T, f ) is lexicographically least. This implies that ρ(T,f) = ρ(G) and that (T , f ) is linked, since we have just proved that decompositions that are not linked are not lexicographically least.

4. Graphs with bounded tree-width

In this section, we prove that the set of graphs having tree-width at most w is well- quasi-ordered. The proof comes in two parts: the first is an argument to provide some structure in a ‘minimal counter-example’ and the second shows that this structure does not occur in graphs with tree-width at most w. Let w be a positive integer and let G be an infinite set of graphs, all with tree- width at most w. For each G ∈ G, we fix, once and for all, a linked decomposition (T , f ) with ρ(T,f) ≤ w and (T ) ≤ 3. We shall assume that there is a leaf r of T such that r/∈ f (E(G)); r is the root of T . (There is no difficulty in arranging for such an r; for example, subdivide any edge of T and join a new vertex r to the subdivision vertex.) We consider the edges of T as being directed towards the root. We note that Xr = ∅ and that, if τ is the edge of T incident with r, then Eτ = E(G). An edge τ of T is labelled with the integer λ(τ) =|(Eτ )|, which is in the set {0, 1,...,w}. For two edges τ,τ of T , τ is linked to τ if T has a directed path P whose first edge is τ, whose last edge is τ , λ(τ) = λ(τ ), and if τ is any other edge of P , then λ(τ ) ≥ λ(τ). Since T is linked, this implies that if τ is linked to τ , then in G there are λ(τ) disjoint paths from (Eτ ) to (Eτ ). 5 Graph minors: generalizing Kuratowski’s theorem 95

More generally, a rooted forest is an infinite collection F of rooted trees, each directed toward its root. Let F be a rooted forest and let λ: E(F) → {0, 1, 2,...,w}. For two edges τ,τ of the component T of F , τ is λ-linked to τ if T has a directed path P whose first edge is τ, whose last edge is τ , λ(τ) = λ(τ ), and if τ is any other edge of P , then λ(τ ) ≥ λ(τ). For a subset A of E(F), the set UF (A) denotes the set of edges τ of F for which there is a directed path in F of length 2 whose first edge is τ and whose second edge is in A. The lemma we need is the following version (taken from [4]) of some of the results in [6]. This is essentially Lemma 2.2 in [6], but the proof seems to be a little tidier.

Lemma 4.1 Let F be a rooted forest and let λ: E(F) →{0, 1,...,w}. Let be a quasi-order on E(F) such that:

(i) there is no inﬁnite descending chain; (ii) if τ is λ-linked to τ , then τ τ.

If the edges of F are not well-quasi-ordered by , then there exists an inﬁnite antichain A of edges of F such that is a well-quasi-order on UF (A).

To see how this applies, let (T , f ) and (T ,f) be two of our decompositions, for G and G, respectively. For τ ∈ E(T ) and τ ∈ E(T ), τ τ means roughly that the subgraph of G induced by Eτ (on the side of τ ‘away from’ r) is a minor of the subgraph of G induced by Eτ (on the side of τ ‘away from’ r ). We also require the minor relation to take the sets (Eτ ) and (Eτ ) into account, but this is a technicality that we ignore for the moment. The lemma says that there is an inﬁnite antichain A of edges of F so that, looking at the tails of the edges of A, the ‘left’ and ‘right’ edges above the tails are well-quasi-ordered. This says that the set consisting of all these subgraphs, both left and right, is well-quasi-ordered, but that the set of graphs (one from each tree) that are the union of the left subgraph and the right subgraph is an inﬁnite antichain. The boundedness of the (Eτ ) will then be used in a straightforward way to obtain a contradiction.

Proof of Lemma 4.1 Suppose that the lemma is false and that, among all counter- examples, we have F and λ for which w is as small as possible. Let Z be the set of edges τ of F such that λ(τ) = 0. Suppose that Z is well-quasi-ordered by . Then the labelling λ − 1onF − Z gives a counter-example, but with smaller w, a contradiction. (It is simple to see that if is a well-quasi-order on both R and S, then it is a well-quasi-order on R ∪ S.) Hence, is not a well-quasi-order on Z; in particular, Z = ∅.Ifτ and τ are in Z, then τ is λ-linked to τ if and only if there is a directed path in F starting with τ and ending with τ . A sequence (τ1,τ2,...) of edges of F is never-increasing if the sequence is inﬁnite and if i

Claim There is a never-increasing sequence (τ1,τ2,...)in Z such that, for each k,ifτk is λ-linked to τ ∈ Z \{τk}, then (τ1,τ2,...,τk−1,τ)is not the beginning of a never-increasing sequence in Z.

To prove this, we note that, since has no infinite descending chain, every non- empty subset of E(F) has a minimal element, and every infinite sequence contains either an infinite antichain or an infinite ascending chain. Note that the assumption that is not a well-quasi-order implies that there is a never-increasing sequence (for example, any ordering of a countably infinite antichain). If we let E1 be the set of edges τ that are the first term of some never-increasing sequence, then we can choose τ1 to be a minimal element of E1. Suppose that we have determined τ1,τ2,...,τk−1; we then select τk to be a minimal element of the set Ek of those τ such that some never-increasing sequence begins with (τ1,τ2,...,τk−1,τ). It is easy to see that (τ1,τ2,...) is a never-increasing sequence. Also, the minimality of τk when it was selected shows that if τ ∈ Z\{τk} is such that τ τk, then (τ1,τ2,...,τk−1,τ)is not the beginning of a never-increasing sequence. Furthermore, suppose i and j are distinct indices such that τj τi. The fact that the sequence is never-increasing implies that i

to Wk) and such that each (Gi+1,Wi+1) is obtained from (Gi,Wi) in one of the following ways:

• by deleting an edge e, in which case Gi+1 = Gi − e and Wi+1 = Wi; • by contracting an edge e = uv, in which case Gi+1 = Gi/e and Wi+1 is Wi,if {u, v}∩Wi = ∅; otherwise, Wi+1 is (Wi \{u, v}) ∪{u }, where u is the vertex of contraction of e; • by deleting an isolated vertex u not in Wi, in which case Gi+1 = Gi − u and Wi+1 = Wi. So now let w be a positive integer and let G be an inﬁnite set of graphs, all with tree-width at most w. For each G ∈ G, let (TG,fG) be a linked decomposition for which ρ(TG,fG) ≤ w and (T ) ≤ 3. We assume that there is a leaf vertex rG of TG not in fG(E(G)), which is the root of TG. Let F be the collection of all trees TG, for G ∈ G. Deﬁne the relation on the edges of F as follows. Let τ ∈ E(F); then there is a Gτ ∈ G such that τ ∈ E(TGτ ).

Let Eτ be that set of edges displayed by τ in (TGτ ,fGτ ), chosen so that f(Eτ ) is not in the component of TGτ − τ containing rGτ . Let Hτ be the subgraph of Gτ induced by Eτ . For edges τ and τ of F , τ τ if (Hτ , (Eτ )) ≤r (Hτ , (Eτ )). The following result is required in order to be able to apply Lemma 4.1. (This is where we use the fact that the decompositions are linked.) Recall that λ(τ) = |(Eτ )|. Lemma 4.2 If τ is λ-linked in F to τ , then τ τ. Proof Since τ is λ-linked to τ , they are both in the same tree TG, for some G ∈ G. Furthermore, λ(τ) = λ(τ ) and, for each edge τ in the shortest path in TG containing both τ and τ , λ(τ ) ≥ λ(τ). Notice that Eτ ⊆ Eτ . The decomposition (TG,fG) is linked, so Menger’s theorem implies that there is a set P of λ(τ) disjoint paths from (Eτ ) to (Eτ ). To see that (Gτ , (Eτ )) ≤r (Gτ , (Eτ )), contract the edges in the paths in P and delete the remaining edges of Eτ \ Eτ . We are ﬁnally ready to prove that graphs with bounded tree-width are well- quasi-ordered.

Proof of Theorem 1.2 Let w, G, (TG,fG) (for each G ∈ G), F and be as described in the paragraphs preceding Lemma 4.2. If the edges of F are well- quasi-ordered by , then we claim that there are G, H ∈ G such that G

Now let τ be an edge of F incident with a leaf t of some TG, other than the root of TG. Then either t/∈ fG(E(G)),ort = fG(e) for some edge e of G.In the former case, Gτ is an empty graph and (Eτ ) = ∅. In the latter case, Gτ is either a loop or a link and (Eτ ) has 0, 1 or 2 vertices. Thus there are, up to isomorphism, only six possibilities for the pair (Gτ , (Eτ )). It follows easily that these edges of F are well-quasi-ordered by . Now apply Lemma 4.1 to F ; note that (i) holds trivially and (ii) holds by Lemma 4.2. The preceding paragraph implies that we can assume that the set A guaranteed in the conclusion of Lemma 4.1 does not contain any edges incident with leaves of F , other than possibly those incident with root vertices. In particular, for each τ ∈ A, we can assume that there are two edges ‘above’τ in UF (A). (The preceding paragraph actually shows only that we can assume that there is at least one edge above τ. Since (TG) ≤ 3, there are at most two such edges. But if there were only one, we could add another one joined to a new leaf that is not the image of any edge of G.) Arbitrarily designate one of these two edges as L(τ) and the other as R(τ) (for ‘left’ and ‘right’). For each τ ∈ A, consider the sets (EL(τ)). Each of these sets has at most w elements, and so infinitely many of them have the same size. Let us consider only these: we use wL for their common size. Since the edges L(τ) are well-quasi-ordered by (this is part of the conclusion of Lemma 4.1), we can find an infinite ascending chain

L(τ1) L(τ2) ... .

Now we repeat the above two paragraphs for the edges R(τ1), R(τ2),....This τ ,τ ,... τ ,τ ,... gives an inﬁnite subsequence i1 i2 of 1 2 such that

• (EL(τ )) wL all ij have the same size ; • (ER(τ )) wR all ij have the same size ; • L(τ ) L(τ ) ... i1 i2 ; • R(τ ) R(τ ) ... i1 i2 . τ ,τ ,... τ ,τ ,... For ease of notation, we relabel the sequence so that i1 i2 is 1 2 . For each i, label the elements of (EL(τi )) with distinct elements from { L, L,...,wL} (E ) 1 2 L and label the elements of R(τi ) with distinct elements from R R R {1 , 2 ,...,wR }. Since L(τ1) L(τ2) ... , we can choose the labellings so that when we recognize Gτi as a minor of Gτj (with i

What we are trying to prove is that (Gτi , (Eτi )) ≤r (Gτj , (Eτj )), for some i

do the graphs GL(τi ) and GR(τi ) glue together to make Gτi ? We need that, for some i

Vertices in (EL(τi )) ∩ (ER(τi )) get both left-labels and right-labels. Since L L L there are only finitely many different subsets of {1 , 2 ,...,wL}, infinitely often it is the same one that gives the left-labels in (EL(τi )) ∩ (ER(τi )). Of these, infinitely often it is the same set of right-labels. Thus, we may relabel (going to this subsequence) in such a way that, for each i, the set of left labels and the set of right labels occurring in (EL(τi )) ∩ (ER(τi )) are always the same.

Notice that (Eτi ) ⊆ (EL(τi )) ∪ (ER(τi )). Thus, every vertex of (Eτi ) has a label: some will have only left-labels, some will have only right-labels, and others will have both. Again, there are only ﬁnitely many possibilities for which labels can appear and for which combinations. Thus, we can assume that

(i) the sizes of the (Eτi ) are all the same;

(ii) the set of left-labels in the sets (Eτi ) is always the same;

(iii) the set of right-labels in the sets (Eτi ) is always the same. Note that (ii) and (iii) combine to show that the set of left-only labels is always the same, and that the set of right-only labels is always the same, because if a vertex has both a left-label and a right-label, then these two labels always go together. Now, for i

(GL(τi ), (EL(τi ))) ≤r (GL(τj ), (EL(τj ))) and

(GR(τi ), (ER(τi ))) ≤r (GR(τi ), (ER(τi ))).

We now show that (Gτi , (Eτi )) ≤r (Gτj , (Eτj )).

The graph Gτi is obtained from GL(τi ) and GR(τi ) by identifying the vertices in

(EL(τi )) ∩ (ER(τi )); this identiﬁcation is the same in both Gτi and Gτj . Thus,

(Gτi , (Eτi )) ≤r (Gτi , (Eτi )), as required.

5. Finding large grids

In this section we prove the second of our main results. Recall from Lemma 2.7 that having bounded tree-width is equivalent to having bounded branch-width. Since we shall prove the result for bounded branch-width, we state it here in that form. Theorem 5.1 For each positive integer k, there is an integer f(k)such that every graph with branch-width at least f(k)has a k-grid minor. In order to state the two main ingredients required to prove this theorem, we need some notation. If A, B ⊆ V (G),an(A, B)-path is a path with one end 100 R. Bruce Richter in A, the other end in B, and which is otherwise disjoint from A ∪ B. Also, for A, B ⊂ V (G), λ(A, B) denotes the maximum size of a set of disjoint (A, B)-paths. To state our ﬁrst ingredient, we introduce a new notion. Let p and q be positive integers. A sextuple (G,A,B,X,Y,P) consisting of a graph G, four disjoint subsets A, B, X, Y of V (G), and a set P of disjoint (A, B)-paths is a (p, q)-path- system if |P|≥p and there are q disjoint (X, Y )-paths. Theorem 5.2 Let k, s and t be given positive integers. Then there exist functions p and q of k, s and t such that, for any (p, q)-path-system (G,A,B,X,Y,P), either G has a k-grid minor, or there is a subset P of P of size s and a set Q of (X, Y )-paths of size t such that P ∪ Q is a set of disjoint paths.

The proof depends on ﬁnding appropriate vertex cuts, which we now introduce. Let W,X,Y be subsets of V (G) and suppose W separates X from Y (that is, there is no (X, Y )-path in G − W). Let EW be the set of edges e of G for which there is a path P in G, containing e, such that P starts at a vertex of X \ W and P ∩ W consists of at most one end of P . The following facts are easily shown:

(EW ) ⊆ W and (EW ) ∪ (W ∩ (X ∪ Y))separates X from Y in G. Given a subset E of E(G), let (X,Y )(E ) denote the union of (E ) with all x ∈ X incident with an edge of E(G) \ E and with all y ∈ Y incident with an edge in E . It is a routine exercise to prove that (X,Y )(E ) separates X from Y . Furthermore, (X,Y )(EW ) ⊆ W. Thus, all minimal sets separating X from Y are of the form (X,Y )(E ), for some E ⊆ E(G). We comment that (X,Y )(E ) is not, in general, the same as (Y,X)(E ) – they differ as to which elements of X and Y are included. Let G be a graph and let E ⊆ E(G). Then (E) is k-connected in G − E if, for any two subsets A and B of (E) with |A|=|B|≤k, there is a set of |A| disjoint (A, B)-paths in G − E, each of which is disjoint from (E) except for its ends. Here is our second ingredient. Theorem 5.3 Let h ≥ k ≥ 2 be positive integers and let G be a graph. Then either G has branch-width at most h or there is a subset E of E(G) such that (i) λ(E) = h; (ii) (E) is k-connected in G − E; (iii) there is a path P such that E(P) ⊆ E and P contracts in G onto (E). (Both of these theorems are essentially as in [3], but the forms we present here seem slightly simpler, with various parts suggested by Jim Geelen and Paul Seymour.) 5 Graph minors: generalizing Kuratowski’s theorem 101

Proof of Theorem 5.2 In this subsection we prove Theorem 5.2. Given k, s and t, let r and q ≥ 2s + t be large enough so that the bipartite Ramsey result holds: any colouring of the edges of the complete bipartite graph Kr,q with two colours (red and blue) either has a K K p = s + ( r + )q (G,A,B,X,Y,P) red s,t or a blue 2k2k+1,k2 . Set 2 2 and let be a (p, q)-path-system. We begin with some simple observations. If G has an edge e not in any path in P, so that there are at least q disjoint (X, Y )-paths in G − e, then we may freely delete e and continue with G − e in place of G. Similarly, consider an edge e in a path P ∈ P such that P has length at least 2. If G/e has at least q disjoint (X, Y )-paths, then we proceed with G/e in place of G. Thus, we may assume that any edge not in any path in P is in every collection of q disjoint (X, Y )-paths in G, and likewise, every edge in a path in P, as long as it is not the only edge in that path, joins vertices in an (X, Y )-separation of size q. If there are at least s paths in P of length 1, then we may use these for P.As long as q ≥ t + 2s, any set of q disjoint (X, Y )-paths contains at least t paths that are totally disjoint from the paths in P. So we may suppose that P has r

Next ﬁx any set Q of q disjoint (X, Y )-paths and let P ={P1,P2,...,P2r }. By the bipartite Ramsey result, either there are s paths in P and t paths in Q that are all disjoint (in which case we are done) or there are a subset P of P of size 2k2k+1 and a subset Q of Q of size k2 so that every path in P intersects every path in Q. P ={P ,P,..., The latter outcome is the only one left to consider. Let 1 2 P } be labelled in increasing order of how they appear in P. In particular, 2k2k+1 P ,P,...,P are all separated from each other by cuts of size q, and they 2 4 2k2k+1 are all disjoint from these cuts. As we proceed from X to Y , every path in Q intersects each of these separating cuts in a single vertex, and therefore intersects the (A, B)-paths in precisely the order P ,P,...,P . 2 4 2k2k+1 To complete the proof, we show that {P ,P,...,P }∪Q contains a 2 4 2k2k+1 k-grid. We begin with an easy fact.

Lemma 5.4 Let ≥ 2 and p ≥ 1 be integers. Let T be a tree with leaves and a longest path of length at most p. Then |V(T)|≤( − 2)(p − 1) + p + 1.

Proof We proceed by induction on , the base = 2 being trivial. If >2, then T has a path P that has one end a leaf, the other end a vertex v of degree at least 3, and all internal vertices having degree 2. Clearly P = P − v has length at most p − 2, T = T − V(P) is a tree with − 1 leaves, and |V(T)|=|V(T)|+|V(P)|.

We are now ready for the ﬁnal argument.

Lemma 5.5 Let k ≥ 2, and let (G,A,B,X,Y,P) be a (k2k+1,k2 −3k +5)-path- system. Suppose Q is a set of k2 − 3k + 5 disjoint (X, Y )-paths such that the paths in P are all edge-disjoint from the paths in Q and every path in Q meets the paths in P in the same order. Then G contains a k-grid minor.

2k+1 Proof Let m = k , and let P ={P1,P2,...,Pm} be such that every path in Q meets the paths in P in the order P1,P2,...,Pm. For each i = 1, 2,...,m, let Ki be the graph whose vertices are the paths in Q and the paths Q and Q in Q are adjacent in Ki if there is a subpath of Pi that has one end in Q, one end in Q and is otherwise disjoint from all the paths in Q. Since Pi intersects all the paths in Q, Ki is connected. Let Ti be a spanning tree of Ki. Since Ti has (k − 2)(k − 1 − 1) + (k − 1) + 2 vertices, Lemma 5.4 implies that either Ti has at least k leaves or Ti has a path of length at least k − 1. Let (Qi,1,Qi,2,...,Qi,k) be a sequence of vertices of Ti which are either all leaves of Ti or (in order) the vertex sequence of a path in Ti. Thus, for each j = 2, 3,...,k, there is a subpath Pi,j of Pi whose ends are in the paths Qi,j−1 and Qi,j and is otherwise disjoint from Qi,1 ∪ Qi,2 ∪···∪Qi,k. There are only s = k2 − 3k + 5 paths in Q, and consequently there are only k 2k s!/(s − k)!≤s ≤ k different possible sequences (Qi,1,Qi,2,...,Qi,k). Thus, 5 Graph minors: generalizing Kuratowski’s theorem 103

k some sequence occurs at least m/s ≥ k times. Let (Q1,Q2,...,Qk) be such a sequence, and let i1

(Qi,1,Qi,2,...,Qi,k) = (Q1,Q2,...,Qk).

The Qm, for m = 1, 2,...,k, will be the ‘horizontal’ paths in the k-grid, while the Pij , for j = 1, 2,...,k, will provide the ‘vertical’ paths in the k-grid.

Speciﬁcally, the vertical paths will be the subpaths of Pij from Qm−1 to Qm, for m = 2, 3,...,k, that are otherwise disjoint from Q1 ∪ Q2 ∪···∪Qk. In order to see exactly the k-grid minor, we contract the portions of Qm, for m = 2, 3,...,k− 1, between the ends in Qm of the segments of Pij joining Qm to Qm−1 and Qm+1. It is here that we use the fact that all the Qm meet the Pij in the same order. The proof of Theorem 5.2 is completed by observing that

|{P ,P,...,P }| = k2k+1 |Q|=k2 ≥ k2 − k + . 2 4 2k2k+1 and 3 5

The second of the following corollaries is what we need.

Corollary 5.6 Let k, n and mi (i = 1, 2,...,r), be positive integers, and let G be X Y A B (i = , ,...,r) V (G) a graph. Let , , i, i 1 2 be pairwise-disjoint subsets of . i = , ,...,r P (A ,B ) r P For 1 2 , let i be a set of disjoint i i -paths, so that i=1 i is a set of disjoint paths. If, for i = 1, 2,...,r, |Pi| and λ(X, Y ) are sufﬁciently large (as a function of k, n and the mi), then either G contains a k-grid minor or i = , ,...,r P ⊆ P Q (X, Y ) there exist, for 1 2 , i i and a set of -paths such that ( i = , ,...,r) |P|≥m |Q|≥n r P ∪ Q for 1 2 , i i, , and i=1 i is a set of disjoint paths. Proof We proceed by induction on r, the case r = 1 being Theorem 5.2. If r>1, Theorem 5.2 implies that either G contains a k-grid minor or there are a large set Q (X, Y ) P ⊆ P |P |≥m P ∪ Q 1 of -paths and a subset 1 1 such that 1 1 and 1 1 is a set of disjoint paths. The result now follows easily by induction.

Corollary 5.7 Let k, mi (i = 1, 2,...,r) and nj (j = 1, 2,...,s) be positive integers, and let G be a graph. Let (Ai,Bi)(i= 1, 2,...,r)and (Xj ,Yj )(j = , ,...,s) V (G) i = , ,...,r P 1 2 be pairwise disjoint subsets of . For 1 2 , let i be (A ,B ) r P |P | a set of i i -paths so that i=1 i is a set of disjoint paths. If the i , for i = 1, 2,...,r, and λ(Xj ,Yj ), for j = 1, 2,...,s, are sufﬁciently large (as a function of k, the mi and the nj ), then either G contains a k-grid minor or there exist Pi ⊆ Pi, for i = 1, 2,...,r, and, for j = 1, 2,...,s, a set Qj of (Xj ,Yj )- ( i = , ,...,r)|P|≥m j = , ,...,s |Q |≥n paths such that for 1 2 i i, for 1 2 , j j , ( r P) ∪ ( s Q ) and i=1 i j=1 j is a set of disjoint paths. 104 R. Bruce Richter

Proof The existence is established by induction on s, the case s = 1 being Corollary 5.6. For s>1, apply Corollary 5.6 to the (Ai,Bi) and (X1,Y1),to k G P ⊆ P ﬁnd either a -grid minor of or very large subsets i i and a very large set Q (X ,Y ) ( r P)∪Q r + 1 of 1 1 -paths, so that i=1 i 1 is disjoint. There are now 1 pairs (Ai,Bi) and only s − 1 pairs (Xj ,Yj ). The result follows.

We note that in Corollary 5.7 we can allow r = 0 (in which case we set (A1,B1) = (X1,Y1) and let P1 be any set of λ(X1,Y1) disjoint paths) and we can allow s = 0 (trivially). We will actually use this result when r = 0.

Proof of Theorem 5.3 We are now ready to prove Theorem 5.3. The correct image for the path P is the following: P = (v0,v1,...,vr ) has its ends v0,vr ∈ (E) and there are 0

Proof We suppose that the branch-width of G is greater than h. Let E∗ be a maximal subset of E(G) such that (a) λ(E∗) ≤ h; ∗ (b) if (T , f ) is a branch decomposition such that E = Eτ for some edge τ of T , then some edge τ in the component of T − τ not containing f(E∗) has ζ(τ)>h; (c) there is a path P in G such that E(P) ⊆ E∗ and P contracts in G onto (E∗). To see that E∗ exists, it sufﬁces to observe that ∅ satisﬁes all three conditions. Moreover, (b) implies that E∗ = E(G). Part 1. λ(E∗) = h. For if λ(E∗)

∗ ∗ λ(E ∪{e}) ≤ 1 + λ(E ) ≤ h, and there is a path P , whose edges are all in E∗ ∪{e}, that contracts onto (E∗ ∪{e}). The contradiction comes from (b). By the maximality of E∗, there is a branch ∗ decomposition (T , f ) of G such that E ∪{e}=Eτ , for some edge τ of T , and every edge τ in the component of T − τ not containing f(e)has ζ(τ) ≤ h. Let T1 and T2 be the components of T − τ, so that f(e)is a vertex of T1. Let (T ,f) be the branch decomposition of G obtained as follows. The vertices of T are those of T , with two new vertices t1,t2 that are adjacent in T . In addition, t2 5 Graph minors: generalizing Kuratowski’s theorem 105 is adjacent to the two vertices of T incident with τ. All other adjacencies in T are in complete agreement with adjacency in T − τ. We let f (e) = t1, and otherwise f (e ) = f(e). Clearly, the label ζ(t1t2) is at ∗ most 2, the edge joining T1 to t2 has label λ(E )

Part 2. (E∗) is k-connected in G − E∗.

If not, there are subsets A, B of (E∗) such that |A|=|B|=s ≤ k, and there is no set of s disjoint (A, B)-paths in G − E∗ with each path disjoint from (E∗) except for its ends. Choose A and B so that s is as small as possible. Let H be the subgraph of G induced by the edges in E(G)\E∗, with the vertices of (E∗) \ (A ∪ B) deleted. Since we need to work in both G and H for the next few paragraphs, we include, as a superscript, the name of the graph (G or H )in which we are determining (F ), for some set F of edges. H Let E ⊆ E(H) be such that S = (A,B)(E ) is a smallest set that separates A from B in H . Clearly, |S|=s − 1 (by the choice of s). By Menger’s theorem, there is a set Q of s − 1 disjoint (A, S)-paths in H. Claim G(E∗ ∪ E) ⊆ (G(E∗) \ A) ∪ S.

Proof Suppose that z ∈ G(E∗ ∪E). Then z is incident with an edge e/¯ ∈ E∗ ∪E and an edge e ∈ E∗ ∪ E.Ife ∈ E∗, then z ∈ G(E∗), while if e ∈ E, then z ∈ H (E) ⊆ S. Thus, G(E∗ ∪ E) ⊆ G(E∗) ∪ S.

Let u ∈ A \ S. Then u is not incident with an edge of E(H)\ E, so every edge in G incident with u is in E∗ ∪ E. Hence u/∈ G(E∗ ∪ E). ∗ It follows immediately from the Claim that λG(E ∪ E )

2 2 components, one isomorphic to TA, one isomorphic to TB , and the third, T3, with ∗ 2 at least |E | leaves that are also leaves of T . Moreover, the edge joining TA to t 2 2 is incident with the end of τA in TA, and similarly for the edge joining TB to t. ∗ Deﬁne f by f(a)= fA(a) for a ∈ E , f(b)= fB (b) for b ∈ E and, for e ∈ E , f(e)is any leaf of T3 that is a leaf of T , chosen so that f is an injection. It is easy ∗ to see that if τ is the edge of T joining T3 to t, then E = Eτ , while if τ is any 2 2 ∗ edge of TA ∪ TB , then ζ(τ ) ≤ h, thereby contradicting the choice of E . It follows that (E∗) is k-connected in E∗.

Proof of Theorem 1.3 We can now obtain the main result of this section, the proof of Theorem 1.3.

Proof of Theorem 1.3 By Corollary 5.7 (with r = 0, s = 2k2 and each ni = 1), there is an integer F(k)such that if G is a graph and (Xj ,Yj ), j = 1, 2,...,2k2, are disjoint subsets of V (G) with the property that λ(Xj ,Yj ) ≥ F(k), for each j, then either G has a k-grid minor or there is, for each j,an(Xj ,Yj )-path Pj such {P ,P ,...,P } that 1 2 2k2 are disjoint. We may suppose that G has branch-width greater than 2k2F(k). By Theorem 5.3, there is a subset E∗ of G such that λ(E∗) = 2k2F(k), (E∗) is (2F(k))- connected in E(G) \ E∗; and there is a path P such that E(P) ⊆ E∗ and P ∗ ∗ contracts in G onto (E ). Let (v1,v2,...,v2F(k)) be the vertices of (E ) in the order in which they appear in the contraction of P . Let V1,V2,...,Vk2 be the partition of (E∗) obtained by putting 2F(k) consecutive vertices into each of these sets. Thinking of the path P as giving us a Hamiltonian path in the k-grid minor that starts and ends in different corners of the grid, we see that we need only connect some of the sets Vi in pairs in order to make a k-grid minor. We will attempt to ∗ join these pairs of Vi using only edges in E(G) \ E , in order not to interfere with the Hamiltonian path. Each Vi is to be joined to at most two others. For each i, let j and j be the indices for which we are trying to join Vi to both Vj and Vj . (Some i have only one such index j.) Arbitrarily partition Vi into two sets Vi,j and Vi,j of size F(k). Consider the pairs (Vi,j ,Vj,i). Note that λ(Vi,j ,Vj,i) ≥ F(k) for each pair i and j. By the definition of F(k), either there is a k-grid minor in G or we can find disjoint paths in E(G) \ E∗ between the required pairs so that we can complete a k-grid minor of G. In either case, G has a k-grid minor. The best result to date for the ‘sufficiently large’ is by Robertson, Seymour 5 and Thomas [8], who proved that a tree-width of order 29k suffices to ensure the existence of a k-grid minor. Not surprisingly, this requires more sophisticated methods than those used here. 5 Graph minors: generalizing Kuratowski’s theorem 107

6. Embedding large grids

In this section we prove Theorem 1.4. The presentation here is based on Geelen, Richter and Salazar [5]. In this section, if C is a set, then C = γ ∈C γ . For a surface S with Euler characteristic ε, the Euler genus of S is µ(S) = 2 − ε(S); this is the number k + 2h, where S is obtained from the sphere by the addition of k crosscaps and h handles. The main topological ingredient we need is the following.

S C Lemma 6.1 Let be a surface and let be a set of pairwise-disjoint simple closed non-contractible curves in S such that some component of S \ C has all of the curves in C in its boundary. Then |C|≤µ(S). Proof We may assume that C is ﬁnite. Let be a component of S \ C for which all the curves of C are contained in the boundary of . Suppose that C contains a separating curve γ , and let S1 and S2 be the components of S \ γ , where is contained in S1. Since γ is non-contractible, S2 is not a disc, and hence contains a non-separating closed curve γ . Replacing γ with γ in C reduces the number of separating curves in C. Thus, we may assume that C contains only non-separating curves. Let c be the number of components of S \ C, let h be the number of 2-sided curves in C, and let k be the number of 1-sided curves in C. Since each curve in C is incident with and each 1-sided curve is incident only with , some collection of h − (c − 1) 2-sided curves in C does not separate S. Each curve in this collection contributes a handle to S. Since the 1-sided curves are pairwise disjoint, each contributes a crosscap to S. Thus, µ(S) ≥ 2(h − (c − 1)) + k. c = µ(S) ≥ h+k ≥ h+k =|C| If 1, then 2 , as required. Thus, we may assume that c>1. Since C contains no separating curve, each component of S \ C other than has at least two curves in its boundary. Each curve in the boundary of such a component is 2-sided and has on the other side, so h ≥ 2(c − 1). Therefore, µ(S) ≥ 2(h − (c − 1)) + k ≥ h + k =|C|, as required.

We need to turn Lemma 6.1 into a form that is useful for graphs. A subgraph H of a graph G separates G if H has more than one bridge in G.

G S C Lemma 6.2 Let be a connected graph embedded in a surface and let be a set of pairwise disjoint cycles in G such that C does not separate G. If every cycle in C is non-contractible in S, then |C|≤µ(S). C G S \ C Proof Since does not separate , there is one component of in which the rest of G is embedded. Since G is connected, each cycle in C is in the boundary of this component. By Lemma 6.1, |C|≤µ(S). 108 R. Bruce Richter

More specifically, we now focus on embeddings of grids. We use Lemma 6.2 to show that if a large grid is embedded in a surface, then some large subgrid is embedded in a disc (Lemma 6.3) and some cylinder that ‘goes around the centre’ is embedded in an annulus (Lemma 6.4). For these results, we require some additional concepts. The boundary cycle of a k-grid G = Pk Pk is the boundary cycle of the infinite face in the usual planar embedding of the grid (that is, it consists of the subgraph induced by the vertices v {i, j}∩{ ,k} = ∅ ≤ i ≤ 1 k R i,j for which 1 ). For 0 2 , define the ring i inductively as follows. The ring R0 is the boundary cycle of the k-grid and, for i ≥ 1, the ring Ri is the boundary cycle of the (k − i)-grid obtained by deleting the vertices of R ,R ,...,R ≤ i ≤ 1 (k − ) J the rings 0 1 i−1. For 1 2 2 , the collar i is the subgraph of G induced by the vertices in Ri−1 ∪ Ri ∪ Ri+1. (The notion of a collar can be extended to any number of consecutive rings, and a generalization of Lemma 6.4 below can be proved by the same argument. However, three consecutive rings are sufficient for our purposes.)

Lemma 6.3 Let t, k and be positive integers such that ≥ t(k + 1), and let G be an -grid. If G is embedded in a surface S with Euler genus at most t2 − 1, then some k-subgrid of G is embedded in a closed disc in S such that the boundary cycle of the k-grid is the boundary of the disc.

Proof Clearly G contains t2 pairwise disjoint k-subgrids and no two vertices from distinct subgrids are adjacent in the grid. By Lemma 6.2, not all of these subgrids can have non-contractible 4-cycles, so one subgrid has only contractible 4-cycles. Each of these bounds a closed disc and the union of these closed discs is the required closed disc.

Lemma 6.4 Let S be a surface of Euler genus µ and let k ≥ 8(µ + 1).IfG is a k-grid embedded in S, then one of the collars in G is embedded in a cylinder of S.

Proof By Lemma 6.2, one of the collars J1,J5,J9,... contains only contractible 4-cycles. For such a collar, each 4-cycle bounds a disc, and the union of these discs is a cylinder.

We are now prepared for the proof of Theorem 1.4. More precisely, we prove the following.

Theorem 6.5 Let S be a surface of Euler genus µ. Let k ≥ µ + 3 (8µ + 9), and let G be a graph containing a k-grid as a minor. If, for each edge e, G − e is embeddable in S, then G embeds in S. 5 Graph minors: generalizing Kuratowski’s theorem 109

Proof Let e be any edge of G. An embedding of G − e in S can be used to obtain an embedding of G in S, the surface obtained from S by adding a handle. Let M be a minimal subgraph of G that contracts to the k-grid Gk. The embedding of M in S contracts (in S ) to an embedding of Gk. Since µ(S) = µ(S) + 2 = µ + 2, Lemma 6.3 implies that some (8µ + 9)- subgrid G of Gk is embedded in a disc in S . Let K be a minimal subgraph of M that contracts to G. Then K is also embedded in a disc. Let e be an edge of K either incident with vm,m if k = 2m − 1 is odd, or in the 4-cycle Rm if k = 2m is even. Let K be a minimal subgraph of K − e that contracts to an (8µ + 8)-grid. Embed G − e in S. By Lemma 6.4, there is a collar Ji of K that is embedded in a cylinder. Let bi−1 be the bridge of Ri in G that contains Ri−1, let bi+1 be the bridge of Ri in G that contains Ri+1, and let B denote the set of all other bridges of Ri.We claim that we can arrange the embeddings of G in S and G − e in S so that if b ∈ B, then b is in the cylinder bounded by Ri ∪ Ri+1 either in both embeddings or in neither embedding. Let B denote the subset of B consisting of those bridges that are in the cylinder bounded by Ri ∪ Ri+1 in one embedding but not in the other. Let OD denote the ‘overlap diagram’ for the bridges in B: its vertices are the bridges in B and two bridges are adjacent in OD if they cannot be simultaneously embedded on the same side of Ri. Since each bridge in B must be embedded in the cylinder bounded by Ri−1 ∪ Ri+1, OD is bipartite. One side of the bipartition corresponds to those bridges that are inside the cylinder bounded by Ri ∪ Ri+1 in one embedding and the other side of the bipartition corresponds to the bridges that are outside in the same embedding. Because we are in the cylinder, we can simply switch the embeddings of the bridges in B in one of the two embeddings, so that each one is either in both cylinders or in neither cylinder. Thus, the subgraphs of G that are contained in the cylinders bounded by Ri ∪ Ri+1 are the same in both embeddings. In order to obtain an embedding of G in S, let 1 be the disc in S bounded by Ri and let 2 be a small closed disc in 1, disjoint from G. Let be the cylinder in S obtained by deleting the interior of 2 from 1. Then is bounded by Ri and some other simple closed curve.After identifying the two copies of Ri, replace the cylinder in S bounded by Ri ∪ Ri+1 with the cylinder . This, together with the cylinder bounded by Ri ∪ Ri−1 and the embedding of Bi−1 in S yields an embedding of G in S.

Acknowledgement I thank Paul Seymour and Jim Geelen for many helpful conversations. 110 R. Bruce Richter

References

1. R. Diestel, Graph Theory (2nd edn.), Springer, 2000. 2. P. Bellenbaum and R. Diestel, Two short proofs concerning tree-decompositions, Combin. Probab. Comput. 11 (2002), 541–547. 3. R. Diestel, K. Y. Gorbunov, T. Jensen and C. Thomassen, Highly connected sets and the excluded grid theorem, J. Combin. Theory (B) 75 (1999), 61–73. 4. J. F. Geelen, A. M. H. Gerards and G. Whittle, Branch-width and well-quasi-ordering in matroids and graphs, J. Combin. Theory (B) 84 (2002), 270–290. 5. J. F. Geelen, R. B. Richter and G. Salazar, Embedding grids in surfaces, European J. Combin. 25 (2004), 785–792. 6. N. Robertson and P. D. Seymour, Graph minors IV.Tree-width and well-quasi-ordering, J. Combin. Theory (B) 48 (1990), 227–254. 7. N. Robertson and P. D. Seymour, Graph Minors XX. Wagner’s Conjecture, J. Combin. Theory (B) 92 (2004), 325–357. 8. N. Robertson, P. D. Seymour and R. Thomas, Quickly excluding a planar graph, J. Combin. Theory (B) 62 (1994), 323–348. 9. R. Thomas, A Menger-like property of tree-width: the finite case, J. Combin. Theory (B) 48 (1990), 67–76. 10. C. Thomassen, A simpler proof of the excluded minor theorem for higher surfaces, J. Combin. Theory (B) 70 (1997), 306–311. 11. C. Thomassen, The Jordan–Schönflies theorem and the classification of surfaces, Amer. Math. Monthly 99 (1992), 116–130. 12. K. Wagner, Graphentheorie, BJ Hochschultaschenbücher 248/248a (1970), 61. 6 Colouring graphs on surfaces

JOAN P. HUTCHINSON

1. Introduction 2. High-end colouring 3. A transition from high-end to low-end colouring 4. Colouring graphs with few colours 5. Girth and chromatic number 6. List-colouring graphs 7. More colouring extensions 8. An open problem References

This chapter surveys results on embedded graphs whose chromatic numbers are as large as possible (depending on the embedding surface) and as small as possible (depending on graph properties). Results often follow the pattern of precolouring parts of the graph and then extending the precolouring. Many results arise as analogues of planar colouring results, and many have parallels in the newer area of list-colouring.

1. Introduction

While recognizing the difﬁculty of the four-colour problem (and displaying the fatal ﬂaw ofA. B. Kempe’s alleged solution), P.J. Heawood turned to the analogous and easier question of colouring graphs embedded on non-planar surfaces. In his 1890 paper he discovered and deduced a remarkable formula [28] giving an upper bound on the number of colours needed for a graph embedded on a surface of Euler characteristic ε<2.

111 112 Joan P. Hutchinson

Theorem 1.1 A graph embeddable on a surface of Euler characteristic ε<2 can be coloured with at most H(ε)colours, where √ H(ε) = 1 + − ε . 2 7 49 24

We denote by g∗ the Euler genus of a surface, g∗ = 2−ε. The orientable surface ∗ Sh, the sphere with h handles, has (orientable) genus h, whereg = 2h√, and graphs S H( − h) = 1 + h + embedded on h can be coloured with at most 2 2 2 7 48 1 colours for h>0. The non-orientable surface Nk, the sphere with k crosscaps, has ∗ non-orientable genus √g = k, and embedded graphs can be coloured with at most H( − k) = 1 + k + k> 2 2 7 24 1 colours, for 0. Heawood demonstrated an embedding of K7 on the torus, but was less precise about embedding complete graphs on other surfaces. He claimed (see [28], p. 335]) that H(ε)is therefore the number of colours necessary and sufficient for a surface … – apart from the verification figure, which we have indeed only given for the case of an anchor ring [torus], but for more highly connected surfaces it will be observed that there are generally contacts enough and to spare for the above number of divisions each to touch each.

The question of determining the best possible colour bound for each surface became known as the Heawood map colour problem. It was subsequently ‘observed’ that Heawood’s bound does give the correct chromatic bound, except for the Klein bottle N2 where only six colours are needed (see Franklin [21]). These observations were carried out over 78 years by many, until in 1968 G. Ringel and J. Youngs ﬁnished the proof that the bounds of Theorem 1.1 are always sharp (except for the Klein bottle). At that time this was arguably the deepest and most intricate result and proof in graph theory.

Theorem 1.2 (Heawood map colour theorem) For every non-planar surface S of Euler characteristic ε, except for the Klein bottle, the complete graph KH(ε) embeds on S, and K6 embeds on the Klein bottle.

Thus the question of determining the maximum chromatic number of graphs on surfaces is completely and satisfactorily solved, and so the subject is ﬁnished with total resolution – or is it? Ungar and Dirac (see [18] and its references) realized that there were more facts to be discovered at the ‘high end’of the chromatic range – that is, they investigated the question of which graphs on a surface of Euler characteristic ε<2 require the full complement of H(ε)colours. In this chapter we consider such ‘high-end’ results – chromatic results on embedded graphs that need many colours – and also ‘low-end’ results on embedded graphs that can be coloured with few (even a constant number of) colours, regardless of the surface on which they lie. The 6 Colouring graphs on surfaces 113 following theorem of Thomassen [57] is one of the most important such results (see also Section 4 and Theorem 6.3). Theorem 1.3 For each non-planar surface S, there is a constant w(S) such that every graph embedded on S with every non-contractible cycle having length at least w(S) can be 5-coloured.

Thus, proving a conjecture of Albertson and Stromquist [10], Thomassen showed that all graphs that are embedded in a locally planar fashion on a surface (that is, with all non-contractible cycles sufﬁciently long) can be coloured using only one more colour than is needed for planar graphs (although Thomassen did not use the four-colour theorem in his proof). We develop some of the motivation for the conjecture of such a 5-colour theorem and look at a variety of instances in which classes of locally planar embedded graphs need only one or two additional colours beyond the number needed for their planar counterparts. Thus we aim to display a spectrum of results, arising from the Heawood map colour theorem, and to show that the work of Heawood, Ringel and Youngs was only the beginning of a colourful area of topological graph theory. The background, history and additional details for the material of this chapter can be found in books by Jensen and Toft [38] and by Mohar and Thomassen [45].

2. High-end colouring

Here is the result of Ungar and Dirac mentioned above and its extensions by Dirac [18]. The proofs are not difﬁcult and follow from average-degree arguments derived from Euler’s formula.

Theorem 2.1 (a) A graph embedded on a non-planar surface S of Euler characteristic ε<2 and requiring H(ε) colours must contain KH(ε), except possibly for the non-orientable surfaces N1,N2 and N3. (b) For ε ≤−3 (except possibly for ε =−4 and − 7),an(H (ε) − 1)-chromatic graph on a surface of Euler characteristic ε must contain either KH(ε)−1 or the join KH(ε)−4 + C5. Part (a) gives a strong statement, very different from the analogue for the plane where 4-chromatic graphs need not contain K4 – consider, for example, an odd wheel. The analogous results for N1 and N3 were ﬁnished in Albertson and Hutchinson [7], where it was also shown that a 6-chromatic graph on N2 need not contain K6. For part (b), the (H (ε) − 1)-chromatic graphs were not characterized for certain values of ε, but more is known now. Thanks to the following theorem of Thomassen [58], 6-chromatic graphs on the torus are now completely understood. If G1 and G2 are two graphs with a shared 114 Joan P. Hutchinson

vertex v0 and given edges v0v1in G1 and v0v2 in G2, the Hajós join is the graph with vertex-set V(G1) ∪ V(G2) and edge-set E(G1) ∪ E(G2) ∪{v1v2}\{v0v1,v0v2}. Let H7 denote the Hajós join of two copies of K4; this is known to be 4-critical. Similarly let H11 denote the Hajós join of two copies of K6, which is 6-critical. 3 The circulant graph Cn denotes the graph on vertices labelled 0, 1, … , n − 1, with vertex i adjacent to vertex j precisely when |i − j|≤3 (mod n).

Theorem 2.2 A 6-chromatic graph on the torus contains K6,C3 + C5, K + H or C3 2 7 11.

Thomassen [62] asked whether the 6-chromatic graphs on the Klein bottle must contain one of the first three of Theorem 2.2 or H11. The answer is no, by the characterization of 6-critical graphs given by Chenette et al. [14]. A graph is k-colour-critical if it cannot be (k − 1)-coloured, but every proper subgraph can be; equivalently, G is k-colour-critical if χ(G) = k and, for each vertex or edge t of G, χ(G−t) = k −1. It is not hard to see that every k-chromatic graph contains a k-colour-critical subgraph, so a search for a characterization of k-chromatic graphs on a surface S entails the determination of a minimal list of k-colour-critical graphs that embed on S. For example, since the odd cycles embed on every surface, these form an infinite list of 3-colour-critical graphs for each surface, although there are many others. Similarly, the odd wheels form an infinite set of 4-colour-critical graphs for each surface. Theorems 2.1 and 2.2 give cases in which these lists are finite for certain colour classes and surfaces. Dirac showed that, for every surface S and for every k ≥ 8, the set of k-colour-critical graphs that embed on S is finite, and Mohar [43] and Thomassen [58] pointed out independently that results of Gallai [22] extend this result to include k = 7. To see why Dirac’s result holds, recall from Euler’s formula that the number m of edges of a graph embedded on a surface of Euler characteristic ε satisfies m ≤ 3(n − ε). Thus the average degree, 2m/n, is bounded above by 6 − 6ε/n, which is less than 6 for ε>0, equals 6 for ε = 0, and is less than 7 for ε<0 and n>−6ε. Now an 8-critical graph cannot have a vertex v of degree less than 7, since the removal of v would allow the graph to be 7-coloured and that 7-colouring could be extended to v. Thus an 8-colour-critical graph on a surface with negative Euler characteristic ε has at most −6ε vertices, and there is only a finite number of such graphs (whether or not they embed on the surface). The results of Gallai [22] give even lower bounds on the number of edges in an embedded 7-colour-critical graph, leading to the finiteness of 7-colour-critical graphs on each surface. In contrast, the case of 6-colour-critical graphs is a demanding one, and in [62] Thomassen showed that for each surface there is only a finite number of such graphs (see Section 6). 6 Colouring graphs on surfaces 115

Dropping to 5-colour-critical graphs, we again find an infinite number of such graphs on each non-planar surface, due to some examples of J. Ballantine and S. Fisk; these are triangulations with exactly two odd-degree vertices that are also adjacent (see [10] and [62]). In summary, the question of whether a natural colour class of embedded graphs is finite or infinite is completely solved – or is it? We’ll see some more variations in Section 6. Another high-end colouring result arises from analogy with the plane. It is easy to see that a plane graph is 2-chromatic if and only if it contains an edge and every facial walk contains an even number of edges, counting multiplicities. In [30] we asked the same question for such evenly embedded graphs on surfaces; these are graphs embedded so that each facial walk has even length, now known also as locally bipartite embedded graphs (see Mohar and Seymour [44]). In general, these graphs need fewer colours than arbitrary embedded graphs, and using a proof similar to that of Heawood we showed the following result [30].

Theorem 2.3 A graph evenly embeddable on a surface of Euler characteristic ε< 2 can be coloured with at most ev(ε) colours, where √ (ε) = 1 + − ε . ev 2 5 25 16

For orientable or non-orientable genus-embeddings √ of a surface of Euler genus g∗ = − ε> 1 + g∗ − . 2 0, this bound becomes 2 5 16 7 Using a proof similar to that of Franklin [21], we showed in [30] that the Klein bottle is again an exception, with its best colour bound being 4, attained by an even embedding of K4 in N2 (see Fig. 1). That paper also showed that the bound of 4 is sharp for the projective plane (g∗ = 1), and that 5 is best for the torus (orientable and g∗ = 2), but the general problem of ﬁnding the best colour bound for a graph embedded with all its faces even-sided was not considered again until the 1980s.

a a a

b b b b

b b

a a a

Fig. 1. K5 evenly embedded on the torus, and K4 evenly embedded on the projective plane and on the Klein bottle 116 Joan P. Hutchinson

For general colourings of graphs on surfaces, triangulations may require the most colours (and so present the most difficulties), since adding diagonals to non- triangular faces may increase the chromatic number of the graph. Similarly, a graph embedded with all of its faces even-sided can have diagonals added to each face until the face is bounded by four sides; such an embedded graph is a quadrangulation. Quadrangular embeddings were of independent interest to Hartsfield and Ringel, for example in [27] and in a personal communication the former reported that for each ε<2 the complete graph on ev(ε) vertices embeds as a quadrangulation of surfaces of Euler characteristic ε, except for the Klein bottle and (surprisingly) the double-torus; the latter fact was established by computer verification. Again, it is not hard to determine which graphs, evenly embedded on a surface of Euler characteristic ε, need ev(ε) colours. Archdeacon et al.[11] showed that for ε ≤−2 and ε = 0, an ev(ε)-chromatic evenly embedded graph on a surface of Euler characteristic ε contains Kev(ε). Also, a 5-chromatic evenly embedded graph on S2 contains K5, and a 4-chromatic evenly embedded graph on N2 contains K4; more extensive results are contained in the work of Král and Thomas [42] and Nakamoto, Negami and Ota [47]. Notice that when Kn has a quadrangular embedding on a non-planar surface, all 3-cycles must be embedded as non-contractible cycles. Fisk and Mohar [20] and Gimbel and Thomassen [23] showed that, for each k ≥ 5, there is only a finite number of k-colour-critical graphs of girth 4 on each fixed surface S. For quadrangulations on a fixed non-planar surface S with all non-contractible cycles of length at least 4 (so that the graphs have girth 4), it is easy to construct infinite families of 2-, 3- and 4-colour-critical graphs (see also Gallai [22] and Král and Thomas [42]), except on the projective plane (see Section 4). Sequentially, the next class of graphs to consider would be those embedded with all faces k-sided, for k ≥ 5.Again, recall the situation in the plane where Grötzsch’s theorem [25] states that a triangle-free planar graph can always be 3-coloured; there is now a whole library of proofs of this result and its generalizations to surfaces. Good surveys are found by Steinberg [53], [54], and the simplest proof to date is by Thomassen [64]. In Section 5 we consider some of the related surface results and find that, with the right hypotheses, these generalizations lead to theorems involving few colours; in other words, these results lead to low-end colouring results.

3. A transition from high-end to low-end colouring

Between 1852, when the four-colour problem was posed, and 1976, when it was solved, there were many attempts to generalize and to solve it. One generalization 6 Colouring graphs on surfaces 117 was insightful. In 1968 P. Erdos˝ (see [13]) and Vizing [66] independently asked about the independence ratio µ of a planar graph, where µ(G) is the size of the largest independent set of G divided by its order. If a graph with n vertices can be k-coloured, then a largest independent set has size at least n/k and so has independence ratio at least 1/k. Erdos˝ and Vizing asked whether every planar G µ(G) ≥ 1 µ(G) ≥ 2 graph has 4 , and Albertson [2] showed that 9 for such graphs G. The problem was not solved except by the proof of the four-colour theorem. Although the Heawood map colour theorem had been solved in 1968,Albertson and Hutchinson in [4] and [5] looked at the corresponding independence ratio question for orientable surfaces and at limiting values of the independence ratio of embedded graphs. Let

U(h) ={µ(G): G embeds on Sh} and L(h) ={limit points of U(h)}. U( ) ⊆ 1 , Thus, by the four-colour theorem, 0 4 1 and it is not hard to see that L( ) = 1 , G = K K 0 4 1 (see [6]). By Theorem 2.1, if H(2−2h) or H(2−2h)−1 and if −1 −1 h ≥ 4, then µ(G)>(H(2 − 2h) − 1) , and so L(h) ⊂ (H (2 − 2h) − 1) , 1 . L(h) ⊂ 1 , From this we concluded that 5 1 . Using a different approach and the four-colour theorem, we were able to show more (see [6]). L(h) = L( ) = 1 , Theorem 3.1 0 4 1 , for all positive h.

Our approach was to show that for h>0 every n-vertex√ triangulation of Sh has a non-contractible non-separating cycle of length at most 2n. Here, and previously in [5], we determined bounds on the length of a shortest non-contractible cycle in an embedded graph. This parameter is called the width, and has since become central in the study of graph embeddings and colourings. For example, Albertson and Stromquist [10] and Thomassen [57] show that graphs embedded with large width have low chromatic number (see below). Answering our questions of [32], Thomassen [56] showed that having width larger than the length of each facial walk indicates that an embedding has minimum genus as well as some planar-like properties; similar results were obtained by Robertson and Vitray [52] using large representativity ( or face-width), which is a lower bound on the number of intersection points of a non-contractible curve on the surface with an embedded graph. Equivalently, face-width is a lower bound on the size of a set of faces whose union (including boundaries) contains a non- contractible cycle; this concept was introduced by Hutchinson in [32]. The idea of width and its influence were sufficient to warrant a chapter, ‘The width of embeddings’, in [45]. In the Graph Minors Project of Robertson and Seymour (see [49] for an overview), face-width played a crucial role for surface results; here we mention a locally planar result from that project that has repercussions throughout 118 Joan P. Hutchinson topological graph theory; their work is one of the deepest and most far-reaching results to date in this area (see [50]). Theorem 3.2 For every non-planar surface S and for every graph H embeddable on S, there is a constant c(S, H) for which any graph embedded on S with face-width at least c(S, H) contracts to H. Often of similar utility (see our work [9], [33] and Thomassen’s [57]) is the determination of a set of planarizing cycles, or a planarizing set, in an embedded graph; this is a set of cycles, or vertices, whose removal leaves a planar√ graph. Our results in [6] showed there to be such a planarizing set of size O(h√ n) for a triangulation of Sh; with Miller [35] we found such a set of size O( hn). These small planarizing sets can lead to colourings with few colours – for example, after we remove a set of planarizing cycles, we can 4-colour the remaining graph; then the cycles, when (say) they have even length and are sufficiently far apart, can be coloured with two additional colours, leading to a 6-colour theorem. (Here, ‘far apart’ means that no two are joined by an edge, so the additional two colours cannot conflict.) Five-colour theorems are also possible – here is an informal description of our approach for orientable surfaces [9]. Suppose that we can find in an embedded graph a set of planarizing non-contractible cycles that are mutually far apart and that are ‘nice’, in that each either has even length or contains a vertex of degree 4 or less. We can then cut the surface along these cycles, remove the cycles, and fill in the cuts with discs, two for each cut since the cuts are two-sided. Finally, we triangulate the resulting discs by adding to each disc a new vertex adjacent to all vertices on the surrounding boundary; these boundary vertices were neighbours of the original cycle. The resulting planar graph can be 4-coloured with colours 1, 2, 3 and 4. Using a fifth colour and the fact that the planarizing cycles are far apart, we can rearrange the 4-colouring so that two vertices that correspond to the same non-contractible cycle and cut receive the same colour. Suppose that the two vertices corresponding to a cycle C are coloured 1 in the resulting 4-colouring. Then their neighbours in the planar graph are coloured 2, 3 and 4, and when we re-sew C into the graph and restore a handle to the surface, we can (almost) colour C with 1 and 5 and colour the final small-degree vertex with whatever colour is not present on its neighbours. This is our colouring technique of [9], [10] and [31], leading to 5-colouring theorems that depend upon the planar four-colour theorem. Thomassen [57] also transforms planarizing cycles into suitably nice ones and achieves a 5-colouring without use of the four-colour theorem. Theorem 3.1 shows that, in a limiting sense, graphs on surfaces act like planar (4-colourable) graphs. Perhaps this result and their subsequent work on the torus [10] inspired Albertson and Stromquist to conjecture that every graph embedded on a surface with sufficiently large width (depending on the Euler genus) could be 6 Colouring graphs on surfaces 119

5-coloured. Large width means that locally the graph is planar, where 4 colours are sufﬁcient, and with an extra colour to vary and patch together the 4-coloured pieces, 5-colouring might be possible. Recall that the graphs that need the full H(ε) range of colours are the complete graphs, always tightly embedded with many non-contractible 3-cycles. In [10] the authors showed that every graph embedded on the torus with width at least 8 can be 5-coloured, and in [58] a corollary of Thomassen’s characterization of 6-chromatic toroidal graphs showed that having width at least 4 implies 5-colourability on the torus. We turn now to Thomassen’s 5-colour theorem for surfaces and to its generalizations and extensions.

4. Colouring graphs with few colours

The first results with few colours come from the work of Dirac, Gallai and Thomassen, mentioned in Section 2. The finiteness of the set of k-colour-critical graphs on a surface S for k ≥ 7 follows from their work and shows that the order of a k-colour-critical graph is bounded, with the bound depending on the Euler genus g∗ of S. Specifically, these results imply that a graph embedded on S can be 6-coloured unless it contains an obstruction on at most 69(g∗ − 2) − 2 vertices. Some early work on colouring with few colours is due to Hutchinson [31] and Fisk and Mohar [20]; in the latter the authors showed, independently of the four- colour theorem, that there is a fixed constant d such that every graph embedded on a surface of Euler characteristic ε<2 with width at least d log ε can be 6-coloured; their proof used an elementary counting argument based on Euler’s formula. Why weren’t similar 4-colour theorems attempted? The Ballantine–Fisk examples, mentioned earlier, can be constructed and embedded with arbitrarily large width, so that these graphs (even with large width) cannot be 4-coloured. Thus, the best possible low-end colouring result is Thomassen’s Theorem 1.3. His proof is complex, but insightful and instructive, is independent of the four- colour theorem, and uses a type of colouring extension result whose repercussions are still being explored and exploited. The result holds for both orientable and non-orientable surfaces, but for convenience, as in his paper, we discuss only the orientable case. (Details on the non-orientable case can be found in [62] and [71].) h+ The final result is that a graph embedded on Sh (h>0) with width at least 214 6 can be 5-coloured; this width bound is not claimed to be sharp. In addition, several of the steps along the way are of interest in their own right. First we add more precision than was given in Section 3. Suppose that a set of disjoint chordless non-contractible cycles C1, C2,...,Ch is a planarizing set for G embedded on Sh,soh ≥ 1. For 1 ≤ i

Ci to Cj , and for 1 ≤ i ≤ h, we define dist(Ci,Ci) to be the minimum length of a path starting with an edge incident with Ci on the right side (given a traversal of Ci and an orientation of the surface) and ending with an edge incident with Ci on the left side; see [45] or [57] for a verification that this is well defined. These distances are crucial in colouring results to ensure that colourings on the different Ci or on two copies of the same Ci are compatible, or can be transformed into compatible colourings. Then the more detailed results of [57] that are useful, for example in our work [33] and [9], include the following.

Theorem 4.1 Given positive integers d and h, a connected triangulation of Sh with width at least 8(d + 1)(2h − 1) contains a set of planarizing chordless cycles C1,C2,...,Ch such that dist(Ci,Cj ) ≥ d for 1 ≤ i ≤ j ≤ h. In [62] Thomassen obtained a similar but stronger and more easily proved result, finding planarizing cylindrical triangulations that are also mutually distant and suitably distant from a given set of vertices. In addition, for base-case work, Thomassen proved a new planar 5-colour theorem, independent even of Euler’s formula. A plane graph is a near- triangulation if each face is a 3-cycle, except possibly the outer face. Thomassen showed that, given a planar near-triangulation with its outer boundary properly 2-coloured, or 3-coloured with at most one use of the third colour, we can extend this colouring to a 5-colouring of the entire graph. By combining these results with a more intricate study, Thomassen found cycles and paths so that cutting along them leaves a triangulation of a planar 4h-gon whose boundary can be 2-coloured. This 2-colouring was then extended to a 5-colouring of the interior graph, and this 5-colouring gave a 5-colouring of the original embedded graph. This triple theme of planarizing cycles, a planar boundary-extension colouring result, and a colouring result for related graphs on surfaces, has been a repeated one in Thomassen’s surface-colouring work. For example, the proof of Theorem 2.2 builds upon the result that for a plane graph G with outer boundary C of length at most 6, any 5-colouring of the induced graph on C extends to a 5-colouring of G, unless one of three obvious obstructions occurs. Thomassen’s extension results often prove useful in surface extension work (see Section 7). A question that naturally arises from Thomassen’s work is the extent to which other planar colouring results, such as the four-colour theorem, generalize to locally planar embedded graphs where possibly one extra colour is included. Is the full range of ev(ε) colours needed for locally planar quadrangulations of a surface of Euler characteristic ε? What role do the width and girth play? Do orientable and non-orientable surfaces always behave in the same way, possibly with a finite number of exceptions such as the Klein bottle or the double-torus? Robinson (see [33]) asked the first question and found an answer for the torus: when a graph is embedded with all faces even-sided and with edges sufficiently 6 Colouring graphs on surfaces 121 short, then the graph can be 3-coloured. Again, ‘short’is defined more naturally for a geometer than for a graph theorist, but his ideas, together with those of Theorem 1.3, led to our next theorem [33].

Theorem 4.2 A graph evenly embedded on an orientable surface Sh(h > 0) with width at least 23h+5 can be 3-coloured. Much of the proof structure followed that of Theorem 1.3, with Robinson’s elegantly simple idea of extending a 2-colouring to a 3-colouring. The hardest step was to establish the existence of planarizing ‘nice’ cycles, particularly suited for quadrangulation reductions, but it was proved that in a quadrangulation of Sh there is a planarizing set of h disjoint chordless non-contractible even-length cycles at mutual distance of at least 4. Then 3-colouring was possible, much as 5-colouring was possible with suitably ‘nice’ planarizing cycles (as described in Section 3). For quadrangulations of surfaces a 3-colour theorem is best possible, since three colours are needed on quadrangulations with non-contractible cycles of odd length. The bounds on width are certainly not best possible; in [11], for example, Archdeacon et al. showed that width 9, as opposed to 28, suffices to ensure 3- colourability for quadrangulations of the torus; we conjectured there that width 6 is the correct bound. In [44] Mohar and Seymour gave constructions and examples to show that no universal constant bound on the width implies the 3-colourability of locally planar evenly embedded graphs. Thus some function of the genus is required. Theorem 4.3 For k>0 and each w>0, there is an unbounded sequence 2 >ε1 >ε2 >...such that, for each i ≥ 1, there is a graph evenly embeddable on the surface of Euler characteristic εi with edge- and face-width at least w and chromatic number at least k. Our proof of Theorem 4.2 is technical and intricate; subsequently we found an ‘easier’ proof [37] for finding the even-length planarizing cycles, but the proof is easier only after one uses the very demanding building blocks of Robertson and Seymour [50]. Our work [33] was carried out for orientable surfaces, and the next step was to determine to what extent and how the same ideas would work for non- orientable surfaces. Most fortunately, the work of Youngs [71] appeared at the same time, showing that projective planar quadrangulations, including those of arbitrarily large width, are always 2-chromatic or 4-chromatic – that is, no 3-colour bound is possible, and surprisingly these graphs are never 3-chromatic. Such bi-modal behaviour was studied, using flow theory and circular colouring, with wide-reaching and deep results by DeVos et al. [16]. On the torus and Klein bottle we can easily construct 3-chromatic graphs, using the Cartesian product of two cycles with at least one of odd length, and 122 Joan P. Hutchinson in [40] Klavžar and Mohar gave 4-chromatic quadrangulations of the Klein bottle of arbitrarily large face-width. Also, Fisk and Mohar [20] showed that graphs of girth 4 on a surface of Euler characteristic ε<2 with width at least d log ε, where d is a constant, can all be 4-coloured. The bound is sharp, since there are quadrangulations requiring at least 4 colours on every non-orientable surface [11] (see also Theorem 4.3). These quadrangulations [11] contain an odd-length cycle whose removal leaves the remaining graph on an orientable surface; such a cycle is called a meridian. In [47] Nakamoto et al. showed that a quadrangulation of a non-orientable surface with sufficiently large face-width is 4-chromatic if and only if it contains a meridian. In [11] we characterized those torus and Klein bottle quadrangulations that require 3 and 4 colours, as follows. Define a diagonal curve for an embedded quadrangulation G to be a curve on the surface that meets G only at vertices, and if a segment of the curve passes through a face {a, b, c, d}, then it joins either a and c,orb and d. A diagonal curve is independent if the vertices at which it meets G form an independent set. Then a non-bipartite quadrangulation of the torus is 5-chromatic if and only if it contains K5, is 3-chromatic if and only if there is a non-contractible independent diagonal curve for G, and is 4-chromatic otherwise. On the Klein bottle, a non-bipartite quadrangulation is 3-chromatic if and only if there is an independent diagonal meridian for the graph, and is 4-chromatic otherwise. These results imply that face-width 9 for a quadrangulation on the torus implies 3-colourability, and face-width 7 for a quadrangulation of the Klein bottle implies 3-colourability, provided that the graph contains an even-length meridian. Subsequently, the locally planar 3-chromatic and 4-chromatic quadrangulations of non-orientable surfaces were characterized. In [47], the characterization of Nakamoto et al. was in terms of non-contractible independent bipartizing curves (curves whose removal leaves a bipartite graph) and meridians. In [44] Mohar and Seymour proved that, for each k>0, there is a number c(k) so that a quadrangulation G embedded on Nk with face-width c(k) is 4-chromatic if and only if G contains disjoint cycles C1,C2,...,Ck that separate k disjoint quadrangulations Q1, Q2,...,Qk on Möbius bands from the rest of Nk so that an odd number of the quadrangulations are non-bipartite. In [47] these two characterizations were shown to be equivalent. There is another ‘easy’planar colouring result that carries over to locally planar graphs on surfaces. Although we asserted in Section 2 that triangulations could be more difficult to colour and could require more colours, this is not the case when all vertices have even degree. In a triangulation, consider a vertex, its neighbours, and the subgraph consisting of edges lying on triangles incident with the vertex. When the central vertex has even degree, that local subgraph can be 3-coloured, and in the plane that local 3-colourability extends to the entire graph. Heawood [29] proved the following result: 6 Colouring graphs on surfaces 123

Theorem 4.4 A planar triangulation can be 3-coloured if and only if each vertex has even degree. This result was noted by both Kempe and Heawood. In fact, there have been many incomplete and incorrect proofs of this result, even though the result is not difﬁcult (see Steinberg [53] and [54]). There are now many proofs of this theorem. For example, it has been shown by M. Król that a planar graph is 3-colourable if and only if it is a subgraph of an even-degree triangulation; we present history, references, alternative proofs and results in [34]. Since (for i ≥ 1) K12 i+3 embeds as an even triangulation of the orientable surface of genus i(12 i − 1), Theorem 4.4 is not immediately extendable. In [15] we began to study the colouring of even triangulations on the torus. We conjectured that an even triangulation with sufﬁciently large width on an orientable surface is 4-colourable. The 4-chromatic quadrangulations of [11] with large width on a non-orientable surface yield 5-chromatic even triangulations on the same surface, since adding a vertex of degree 4 to each face of a quadrangulation produces an even triangulation of the same surface. If a 4-colouring of the original quadrangulation necessarilycontainsafacereceivingall4colours,thentherelatedeventriangulation is 5-chromatic. The following results come from [11], [23], [37] and [71].

Theorem 4.5 For each non-orientable surface Nk (k > 0), there are quadrangulations of arbitrarily large width such that, in any colouring of the graph, there is a face receiving four different colours.

Corollary For each non-orientable surface Nk (k > 0), there are even triangulations of arbitrarily large width that require five colours. Nakamoto [46] has characterized these 5-chromatic graphs on non-orientable surfaces; Theorem 4.6 shows that the set is empty for orientable surfaces. A3-colour theorem is not possible for even triangulations of orientable surfaces. Notice that a 3-colouring of a triangulation gives a 2-colouring of the dual embedded graph – each face can be coloured with a ‘clockwise’ or ‘anticlockwise’ colouring, according to the representation of {1, 2, 3} on its boundary. Since even triangulations on surfaces need not have a bipartite dual, they cannot always be 3- coloured; we showed that the conjectured bound of four colours is correct, though, for orientable surfaces (see [37]). Theorem 4.6 Given h>0, there is a constant c(h) such that, if G is an even triangulation of the orientable surface of genus h embedded with all non-contractible cycles of length at least c(h), then G can be 4-coloured. The bound c(h) is not specified and comes from deep results of Robertson and Seymour [50]. In [37] we asked whether there could be a fixed bound on width giving 4-colourability for all surfaces; however, the examples of Mohar and 124 Joan P. Hutchinson

Seymour [44] showed that the width bound must depend on the genus – take their examples of evenly embedded graphs, add a vertex to each face, and join it to each vertex on the boundary of that face. These become even triangulations with arbitrarily large width and chromatic number and with decreasing Euler characteristic.

5. Girth and chromatic number

So far we have considered all graphs but with special embeddings – for example, even embeddings or those with only long non-contractible cycles; we also specialized to even degrees for triangulations. Now we turn to the girth parameter, a purely graph-theoretic one, to see its effect on the chromatic numbers of embedded graphs. One underlying result is that, for each pair of natural numbers j ≥ 2 and k ≥ 2, there exist k-chromatic graphs of girth j; see [13] and [38]. Often these graphs give enlightening examples when considered on surfaces, as in Theorem 4.3. Each of these graphs has its own orientable and non-orientable genus, but from the converse point of view of surfaces it is instructive to ask about the chromatic number of graphs of girth j that embed on a fixed surface S. Notice that the girth gives a bound on face size and on width, but not conversely. The motivating planar result is Grötzsch’s theorem, a non-trivial planar result, as described at the end of Section 2. Grötzsch [25] proved more, adding to the theory of colouring extensions. Theorem 5.1 (Grötzsch’s theorem) Every planar triangle-free graph can be 3-coloured. Furthermore, every 3-colouring of a 4-cycle or 5-cycle can be extended to a 3-colouring of the graph. In his proof of Grötzsch’s theorem, Thomassen [59] gave the following improvement. Theorem 5.2 Let C be a facial cycle of length at most 9 in a planar graph G of girth 5. Then any 3-colouring of the induced subgraph on the vertices of C can be extended to a 3-colouring of G , except when C has length 9 and there is a vertex not on C that is adjacent to three different colours on C. Notice that for an inductive proof of Theorem 5.1, one can assume that G is 2-connected and has minimum degree at least 3, and that a 5-cycle C is 3-coloured initially. If there is a cycle C of length 4 in G, then induction can be applied if C is a separating cycle, or after identifying a pair of opposite vertices if C is a facial cycle. Thus girth 5, as in Theorem 5.2, presents the most challenges. Thomassen has since found an easier proof [64], again by proving more (see Theorem 6.5). To what extent does Grötzsch’s theorem (or a similar one) hold for triangle- free graphs on surfaces? The torus is now well understood, since Thomassen [60] 6 Colouring graphs on surfaces 125 has shown that a toroidal graph of girth 5 can be 3-coloured; in fact, the same colouring bound holds for toroidal graphs in which all contractible cycles have length at least 5. Girth 5 is best possible for 3-colourability, since Gallai [22] proved that there are infinitely many triangle-free 4-chromatic graphs on every surface. In [23] Gimbel and Thomassen asked for a characterization of girth-4 graphs on the torus that are not 3-colourable (since answered by Král and Thomas [42]). They asked whether, on the double-torus, graphs of girth 4 are 4-colourable and those of girth 5 are 3-colourable; they also showed that girth 6 on the double-torus implies 3-colourability. Gimbel and Thomassen’s most general result is that there are positive constants c1 and c2 such that girth-4 graphs embeddable on Sg need 1/3 1/3 at most c1g /(log g) colours, and there are examples with chromatic number 1/3 2/3 at least c2g /(log g) , with related results for embedded graphs of girth j>4 (see [23], [24]). Graphs on low-genus non-orientable surfaces have presented different challenges and results. Answering a question of Woodburn, Thomassen [59] showed that a graph on the projective plane without contractible 3- or 4-cycles can be 3-coloured, so that again girth 5 implies 3-colourability; this result follows easily from Theorem 5.2, and the examples of Youngs [71] provide 4-chromatic graphs of girth 4 on the projective plane. In [23] a 3-colouring extension variation of Theorem 5.2 for girth-4 planar graphs allowed Gimbel and Thomassen to prove a conjecture of Youngs that a graph on the projective plane with every contractible cycle with length at least 4 can be 3-coloured if and only if it does not contain a non-bipartite quadrangulation. On the Klein bottle it is possible to embed K4 with all 3-cycles and 4-cycles non-contractible, so that Thomassen’s toroidal 3-chromatic result does not hold exactly there. In [23] the authors asked whether girth 5 on the Klein bottle implies 3-colourability: this was answered in the affirmative by Thomas and Walls [55], who showed that the embedding of K4, described above, is the first in an infinite family of graphs on the Klein bottle with width 5 that are not 3-colourable. They also showed that, excepting this family, a graph on the Klein bottle with no non- contractible 3-cycle or 4-cycle is 3-colourable, and they asked whether, for each orientable surface Sh (h>0), there is a constant c(h) so that a graph of girth 4 on Sh with width at least c(h) is 3-colourable. By results of [11], [16], [44], [47] and [71], this does not hold for non-orientable surfaces. More results on girth are included at the end of Section 6, since they were proved using list-colourings.

6. List-colouring graphs

Another especially insightful and helpful variation on graph colouring is that known as ‘list-colouring’: this is a natural generalization that was developed independently by Vizing [67] and Erdos˝ et al. [19]. Suppose that to each vertex v 126 Joan P. Hutchinson of a graph G there is assigned a list L(v) of colours that are suitable for colouring v. G is list-colourable if each vertex can be assigned a colour from its list so that adjacent vertices receive different colours, and is k-list-colourable (or k-choosable) if it is list-colourable whenever each list has length at least k. A graph that is k-list-colourable is k-colourable, as can be seen by taking each list L(v) to be {1, 2, ..., k}, but not conversely. For example, for each k>2 there are bipartite non-k-list-colourable graphs (see [19]). A graph is k-degenerate if each of its subgraphs (including itself) has a vertex of degree at most k; such graphs are (k +1)-colourable and (k +1)-list-colourable. Since every planar graph is 5-degenerate, these are 6-list colourable. A theorem with a remarkably elegant proof is the next result of Thomassen [59]; in fact, its proof is so elegant and simple that it is included in Proofs from the Book [1], where in the spirit of Erdos,˝ many of the ‘best’ proofs are collected.

Theorem 6.1 Every planar graph is 5-list-colourable.

Again, Thomassen’s proof involves a colouring extension result and a stronger result than simple 5-list-colourability. Let G be a near-triangulation with outer boundary C. If two consecutive vertices of C are coloured, if lists for all other vertices of C have size 3, and if lists for all remaining vertices have size 5, then the colouring of the initial two vertices extends to a list-colouring of G. This approach was reﬁned in [61] and applied to planar graphs of girth 5. Suppose that G is a planar graph of girth 5 with C a facial 5-cycle and with L(v) ≥ 3 for each vertex v. Then any 3-list-colouring of C extends to a 3-list-colouring of G, giving a stronger result than that of Theorem 5.1. Again the plane is anomalous with a larger list-chromatic number than the chromatic number; Voigt [68] showed that 4-lists are not sufﬁcient for planar graphs. For every non-planar surface, the best colouring bound of Theorem 1.1 also gives the best list-chromatic bound for graphs on that surface (with 6 as the correct bound for the Klein bottle), since a minimal k-list-colourable graph must have minimum degree at most k − 1 (see [38], Section 1.9). Thomassen’s work on list-colouring [62] also led to a solution of the 6-colour- critical problem, mentioned in Section 2; again, more was proved to obtain this result, including a list-colouring result and an extension result. A graph is list- critical if it is not list-colourable, but every proper subgraph is.

Theorem 6.2 For each surface S, there are only ﬁnitely many list-critical graphs on S with every list of size 6 or more.

Again, the building blocks for the proof of Theorem 6.2 contain interesting extension results. The result itself also ﬁnally answers the k-colour-critical question, except when the girth parameter is also considered. Kawarabayashi and 6 Colouring graphs on surfaces 127

Mohar (personal communication) have shown that each surface contains only a finite number of list-critical graphs with lists all of size 5 or more. Further list-colouring results for graphs on surfaces have recently been proved, often giving results parallel to those for colouring and usually with more demanding proofs. For example, the result analogous to Theorem 1.3 is the following striking result of DeVos et al. [17]; the necessary embedding width depends upon the surface. Theorem 6.3 Every graph embedded on a non-planar surface with sufficiently large width (depending on the surface) can be 5-list-coloured. Analogous to Theorem 2.1 is the following result of Böhme, Mohar and Stiebitz [12] and of Král and Škrekovski [41]: Theorem 6.4 For all ε<2, a graph that embeds on a surface of Euler characteristic ε is H(ε)-choosable, and is (H(ε) − 1)-choosable unless it contains KH(ε). Grötzsch’s theorem has also been generalized by Thomassen in [61] and [63]: Theorem 6.5 Every planar graph of girth 5 is 3-list-colourable. This theorem was also proved using a colouring extension from a short path or cycle on the outer boundary. Voigt [69] found a triangle-free planar graph that is not 3-list-colourable. It is not known whether toroidal graphs of girth 5 are 3-list colourable (see [61]). Kawarabayashi and Mohar (personal communication) have shown that each surface contains only a finite number of list-critical graphs with lists all of size 5 or more. Returning to (regular) colourings, we see thatTheorem 6.2 concludes the colour- critical question for arbitrary graphs (that is, for those of girth 3). Is the set of k-colour-critical graphs still finite when embedded graphs of girth 4, 5, 6 or higher are considered? Again, the odd cycles give us infinitely many 3-colour-critical graphs of increasing girth. Using Euler’s formula and results of Gallai [22], we find that there are only finitely many k-colour-critical embedded graphs of girth 6, when k ≥ 4. So are there also only finitely many k-colour-critical embedded graphs of girths 4 and 5, when k ≥ 5? (See Fisk and Mohar [20] and Gimbel and Thomassen [23].) The final question of girth 5 with k = 4 has recently been settled by Thomassen [63], showing again the finiteness of this set of graphs.

7. More colouring extensions

In an early version of [62], Thomassen asked about a more general extension result. If a planar graph G has a set P of independent vertices, precoloured with ﬁve colours, does this colouring extend to a 5-colouring of G provided that the 128 Joan P. Hutchinson vertices of P are suitably far apart? The examples of uniquely 4-colourable planar graphs (for example, planar 3-trees) show that no 4-colour extension theorem for 4-precolourings is possible. In [3] Albertson showed that the answer to Thomassen’s question is yes, and that planarity is not even needed for the following elegantly proved result:

Theorem 7.1 If G is an r-chromatic graph, and if the distance between any two vertices of P is at least 4, then any (r + 1)-colouring of P can be extended to an (r + 1)-colouring of G .

Proof Let c be an r-colouring of G and let d be the precolouring of P .If d(v) = c(v) for some vertex v, then we make no change. If d(v) = r + 1, we can immediately change to that colour at v; otherwise d(v) = c(v) and c(v) ≤ r.We change each neighbour of v that is coloured d(v) to the colour r + 1, and we can recolour v to receive d(v). Since no two vertices of P are at distance less than 4, this recolouring is valid.

The bounds of 4 and r + 1 were shown to be sharp. In many other cases, embeddings of graphs do help with the existence of, and bounds for, colouring- extension theorems. In [9] we considered precoloured vertices in locally planar graphs on orientable surfaces, and showed instances where the precolouring extends to a 5-colouring of the entire graph. In [8] we showed that for graphs on a surface of Euler characteristic ε<2, any H(ε)-colouring of a set of vertices at distance 6 apart extends to an H(ε)-colouring of the entire graph. Once Albertson had swiftly solved Thomassen’s question, Thomassen [62] substituted the following question. If in a planar graph G some bipartite subgraphs that are mutually far apart are each coloured with two of ﬁve colours (but not necessarily the same two colours), does this colouring extend to a 5-colouring of G? Here a topological assumption is essential for obtaining the best results (see [8] and [36]). The answer is yes for planar (and for K5-minor-free) graphs, provided that the bipartite components are at distance at least 8 from one another; furthermore, in Kr+1-minor-free graphs, which are r-colourable for r ≤ 5 (see Section 8), if mutually separated subgraphs are each s-coloured (s ≥ 3) with colours chosen from r +s −1 colours, then the colouring of the subgraphs extends to an (r + s − 1)-colouring of the entire graph. Similarly, on surfaces of Euler characteristic ε<2, an (H (ε) + s − 1)-colouring of subgraphs, with components mutually far apart and each component s-coloured, extends to the entire graph when s ≥ 2. In contrast, without embeddings we see that if G is r-colourable, and if P induces s-colourable subgraphs of G whose components are at mutual distance at least 4, then any (r + s)-colouring of the subgraphs in which each component is s-coloured extends to an (r + s)-colouring of G (see [8]). The total 6 Colouring graphs on surfaces 129 number of colours needed is best possible, and in [36] we sought, and in many cases established, sharp distance constraints. One could argue that (almost) all colouring results are colouring extension results – even a proof by induction shows that a colouring of a subgraph extends to the entire graph. In the work of this section, colouring extensions are considered for their own sake, and not just as stepping stones towards other results.

8. An open problem

One additional historic and on-going colouring problem is that of Hadwiger’s conjecture. In 1943 perhaps in part to remove colouring questions from surfaces, Hadwiger [26] conjectured that every k-chromatic graph contracts to a complete graph on k vertices, and proved the conjecture for k = 4. Many consider this to be one of the hardest open questions in colouring theory. Wagner [70] showed that for k = 5 the conjecture is equivalent to the (now proved) four-colour theorem, and in early work other cases of Hadwiger’s conjecture were proved – for example, for some classes of graphs on surfaces. The case k = 6 has also been shown to be equivalent to the four-colour theorem, by Robertson, Seymour and Thomas [51]. In addition, they have a low-degree polynomial-time algorithm that, for fixed k and input of a graph G, finds either a k-colouring of G, or shows that G contracts to Kk+1, or finds a counter-example to Hadwiger’s conjecture. Two striking and recent results on this conjecture were found by Kawarabayashi and Toft [39] and by Reed and Seymour [48], but the general conjecture remains obstinately open, as explained by Toft [65].

Acknowledgement

This chapter is dedicated to Michael O. Albertson and Herbert S. Wilf, with thanks for their support and for their many examples of excellent writing and mathematics.

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R. BRUCE RICHTER and G. SALAZAR

1. Introduction 2. What is the crossing number? 3. General bounds 4. Applications to geometry 5. Crossing-critical graphs 6. Other families of graphs 7. Algorithmic questions 8. Drawings in other surfaces 9. Conclusion References

The crossing number of a graph G is the smallest number of pairwise crossings of edges among all drawings of G in the plane. In the last decade, there has been signiﬁcant progress on a true theory of crossing numbers. There are now many theorems on the crossing number of a general graph and the structure of crossing-critical graphs, whereas in the past, most results were about the crossing numbers of either individual graphs or the members of special families of graphs. This chapter highlights these recent advances and some of the open questions that they suggest.

1. Introduction

Historically, the study of crossing numbers has mainly been devoted to the computation of the crossing numbers of particular families of graphs. Given that we still do not know the crossing numbers for basic graphs such as the complete graphs and complete bipartite graphs, this is perhaps not surprising. However, a broader theory has recently begun to emerge. This theory has been used for

133 134 R. Bruce Richter and G. Salazar computing crossing numbers of particular graphs, but has also promulgated open questions of its own. One of the aims of this chapter is to highlight some of these recent theoretical developments. The study of crossing numbers began during the Second World War with Paul Turán. In [58], he tells the story of working in a brickyard and wondering about how to design an efﬁcient rail system from the ‘kilns’ to the ‘storage yards’. For each kiln and each storage yard, there was a track directly connecting them. The problem he considered was how to lay the rails to reduce the number of crossings, where the cars tended to fall off the tracks, requiring the workers to reload the bricks onto the cars. This is the problem of ﬁnding the crossing number of the complete bipartite graph. It is also natural to try to compute the crossing number of the complete graph. To date, there are only conjectures for the crossing numbers of these graphs (Fig. 1 suggests how the Kr,s conjecture arises): (K ) = 1 r 1 (r − ) 1 s 1 (s − ) cr r,s 2 2 1 2 2 1 and (K ) = 1 1 n 1 (n − ) 1 (n − ) 1 (n − ) . cr n 4 2 2 1 2 2 2 3

The former is known to be true for r ≤ 6 and all s, and also for r = 7 and 8 when s ≤ 10. The latter is known for n ≤ 12.

Fig. 1.

A third family has played an important role in stimulating the development of some of the theory. The Cartesian product of two cycles makes a toroidal grid. The fact that its crossing number grows with the sizes of the cycles shows that toroidal graphs can have arbitrarily large crossing numbers. The conjecture (see Fig. 2) is 7 Crossing numbers 135

Fig. 2.

that cr(Cr Cs) = (r − 2)s, for 3 ≤ r ≤ s. This has been proved by traditional methods for r ≤ 7. The pioneering work of Beineke and Ringeisen [45], [4] for the cases r = 3 and r = 4 stimulated the work that leads to much of the theory outlined in Section 6. Webegin in Section 2 by introducing four different crossing numbers.The rest of this article is about the standard crossing number. Section 3 provides some general lower bounds for the crossing number, while Section 4 gives one of several lovely applications of these bounds to combinatorial geometry. Section 5 introduces the recently developed structure found in crossing-critical graphs. Section 6 describes some theory developed for us to be better able to determine the crossing numbers of certain families of graphs – this has been especially successful for the Cartesian product of cycles. Some algorithmic questions are discussed in Section 7 and we conclude in Section 8 with a brief look at recent work demonstrating connections between drawings of the same graph on different surfaces. There is an impressive bibliography of crossing number papers maintained by Vrt’o [61].

2. What is the crossing number?

Pach and Tóth [36] ask: what is the ‘right’ deﬁnition of a crossing number? A drawing of a graph G in the plane is a set of distinct points in the plane, one for each vertex of G, and a collection of simple open arcs, one for each edge of the graph, such that if e is an edge of G with ends v and w, then the closure (in the plane) of the arc α representing e consists precisely of α and the two points representing v and w. We further require that no edge-arc intersects any vertex point. We may now try to count crossings of the arcs. We follow Székely’s lead [57] and give four different ways of doing this. Let D be a drawing of a graph G. The 136 R. Bruce Richter and G. Salazar

(standard) crossing number cr(D) of D is the number of pairs (x, {α, β}), where x is a point of the plane and α and β are arcs of D representing distinct edges of G such that x ∈ α ∩ β. The pair crossing number pcr(D) of D is the number of pairs of arcs α and β of D representing distinct edges of G such that α ∩ β = ∅. The point here is that two edges may intersect many times, but contribute only one to pcr(D). The odd crossing number ocr(D) of D is the number of pairs α and β of arcs of D representing distinct edges of G such that |α ∩ β| is odd. The independent odd crossing number iocr(D) of D is the number of pairs α and β of arcs of D representing distinct edges of G that are not incident with a common vertex and such that |α ∩ β| is odd. It is very easy to see that, for any drawing D of G,

iocr(D) ≤ ocr(D) ≤ pcr(D) ≤ cr(D).

For each of these crossing numbers, the corresponding crossing number of G is deﬁned to be the minimum over all drawings of G. For example, pcr(G) = min{pcr(D)}, where the minimum is over all drawings D of G. Thus,

iocr(G) ≤ ocr(G) ≤ pcr(G) ≤ cr(G).

In particular, Pach and Tóth ask:

Problem Is it true that, for all graphs G, pcr(G) = cr(G)?

The independent odd crossing number was introduced byTutte [59], who proved that if iocr(G) = 0, then cr(G) = 0. He also proved the nice property that if D and D are two drawings of a graph G, then iocr(D) ≡ iocr(D)(mod 2). In that paper, Tutte wonders whether iocr(G) is equal to cr(G), for all graphs G. Pach and Tóth [36] proved the following interesting result. Theorem 2.1 cr(G) ≤ 2(ocr(G))2. The proof consists of the non-trivial fact that, given a drawing D of G, there is a second drawing D of G such that if an edge e of G has a crossing in D then, in D, e has an odd number of crossings with some other edge: see Pelsmajer, Schaefer and Štefankovicˇ [39] for a simpler proof and related results. An important recent work [40] by Pelsmajer, Schaefer and Štefankovicˇ gives the ﬁrst example of a graph G for which ocr(G) < cr(G): this answers Tutte’s question in the negative.√ For each ε>0, they show√ that there is a graph G such (G)<(1 + ε) (G) 1 that ocr 2 3 pcr . It may be that 2 3 is the smallest possible coefﬁcient that can occur here, but this is open. Pach’s question, ‘Is pcr(G) = cr(G)?’ is still open. 7 Crossing numbers 137

3. General bounds

Erdos˝ and Guy [15] conjectured the following, which has been proved by Ajtai, Chvátal, Newborn and Szemerédi [2] and independently by Leighton [31]. Theorem 3.1 There exists a positive constant c such that, if m ≥ 4n, then cr(G) ≥ cm3/n2. The proof is an easy induction and generalizes the result slightly to allow m ≥ (3 + ε)n, for any ε>0, but then c becomes a function of ε. Note that this result already implies that cr(Kn) ≥ cn4, for some constant c>0. Pach, Spencer and Tóth [34] generalized this bound as follows. Theorem 3.2 Let g be a positive integer. There is a positive constant c = c(g) such that if G has girth at least 2g + 1 and m ≥ 4n, then cr(G) ≥ cm2+g/n1+g. A graph property is a set of graphs. A graph property P is monotone if • G ∈ P and H ⊆ G implies that H ∈ P; • G1 ∈ P and G2 ∈ P implies that the disjoint union G1 ∪ G2 is in P. For a monotone property P,ex(n, P) denotes the maximum number of edges in a graph G with n vertices such that G has property P. The following general statement is in [34]. Theorem 3.3 Let P be a monotone graph property and suppose that there are positive constants α, c1 and c2 for which

1+α 1+α c1n ≤ ex(n, P) ≤ c2n .

Then there are positive constants c3 and c4 such that, if G ∈ P and G has n 2 vertices and m ≥ c3n log n edges, then m2+1/α cr(G) ≥ c . 4 n1+1/α The conclusion of Theorem 3.1 is the same as that of Theorem 3.3 in the case α = 1. However, notice the hypothesis in the latter requires that m ≥ cn log2 n, rather than m ≥ cn as in Theorem 3.1.

Problem Is Theorem 3.3 still true if the hypothesis is weakened to m ≥ cn?

One sticking point in the proof of Theorem 3.3 given in [34] is ﬁnding a good estimate on the sum of the squares of the degrees of the vertices in a graph in P with n vertices – the trivial estimate +α deg2(v) ≤ n deg(v) ≤ n2 v∈V (G) v∈V 138 R. Bruce Richter and G. Salazar

2 is used. Füredi and Kündgen [19] have shown that v∈V (G) deg (v) ≤ f (n, α), where   n2 if α<1 ,  2 f (n, α) = n2 n α = 1 ,  log if 2  n1+2α 1 <α≤ . if 2 1

Unfortunately, this improved estimate improves the hypothesis of Theorem 3.3 m ≥ cn n α<1 m ≥ cn( n)3/2 α = 1 only to log , for 2 , and to log , for 2 . The accuracy of the estimates in [19] suggests that other methods may be needed to improve the hypothesis to m ≥ cn. Leighton introduced two other important ideas in [31]. One way to estimate the crossing number of a graph G is to immerse G into another graph H .Animmersion of a graph G into a graph H is a mapping of the vertices of G to vertices of H and a mapping of the edges of G to paths in H:ife = vw ∈ E(G), then e is mapped to a path in H whose ends are the images of v and w. There is no requirement that the edge-representing paths be disjoint, but no edge-representing path should have a vertex-representing vertex in its interior. For each edge e of H , we call the congestion of e the number of paths of H corresponding to edges of G that pass through e. The crossing number of G can be readily approximated by a function of the crossing number of H and the maximum congestion of the immersion. This idea has been used by Shahrokhi, Székely and Vrt’o [53] to show that the n-cube has crossing number at least c4n, for some constant c. The other of Leighton’s ideas in [31] we mention here is that of relating the crossing number of a graph G to its bisection width b(G), which is the smallest number of edges in a cutset dividing V (G) into two sets, each with at least a third of the vertices of G. One form of this theorem is the following [33]:

Theorem 3.4 Let G be a graph with n vertices with degrees d , d , ..., dn. Then √ 1 2 b(G) ≤ (G) + n d 2 10 cr 2 i=1 i .

This type of result is used in the proof of Theorem 3.3 and shows why estimates 2 on di are wanted. In a slightly different direction, Pach and Tóth [35] considered a graph having a drawing in the plane for which no edge crosses more than k other edges. This obviously restricts the number of edges and they proved that, for k ≤ 4, m ≤ (k + 3)(n − 2) (generalizing the m ≤ 3n − 6 bound for planar√ graphs) while, for general k, there is a universal constant c such that m ≤ c kn. 7 Crossing numbers 139

4. Applications to geometry

In a beautiful paper, Székely [56] has shown how to use Theorem 3.1 to provide simple proofs of ‘hard’problems in combinatorial geometry. The simplest example is the Erdos-Trotter˝ theorem:

Theorem 4.1 Given n points and lines in the plane, there is a constant c for which the number of incidences among the points and lines is at most c((n)2/3 + n + ).

Proof Let G be the graph with the n points as vertices, in which two vertices are adjacent if they are consecutive points on one of the lines. The arrangement of points and lines gives a drawing of G in the plane. Since no two lines cross more than once, we have cr(G) ≤ 2. On the other hand, in each line the number of vertices is one greater than the number of edges, so the number I of point- line incidences is at most m + . Therefore, by Theorem 3.1, either 4n ≥ m or cr(G) ≥ cm3/n2 – that is, either I ≤ 4n + or 2 ≥ c(I − )3/n2.

5. Crossing-critical graphs

A graph G with crossing number k or more is k-crossing-critical if every proper subgraph has crossing number less than k, and G is homeomorphically minimal. Note that this last condition, together with Kuratowski’s theorem, implies that the only 1-crossing-critical graphs are K3,3 and K5. We remark that there are graphs G (examples are mentioned below) that are k-crossing-critical for some k

Theorem 5.1 If G is 2-crossing-critical, then either cr(G) = 2 or G = C3 C3.

Since cr(C3 C3) = 3, and so not every 2-crossing-critical graph has crossing number 2, Theorem 5.1 can be stated in a less precise way: there is a constant c such that if G is 2-crossing-critical, then cr(G) ≤ c. This suggests the following result, proved by Richter and Thomassen [44]: G k (G) ≤ 5 k + Theorem 5.2 If is -crossing-critical, then cr 2 16. Lomeli and Salazar [32] have shown that there is a constant c such that, if G is sufficiently large and k-crossing-critical, then cr(G) ≤ 2k + c. The important point is that the crossing number of a k-crossing-critical graph is bounded by a function of k.Ifn ≥ 5, then the complete graph Kn (for example) is k-crossing-critical for some (unknown) k. Since k is of the order of n4, some 140 R. Bruce Richter and G. Salazar edge must cross about n2 edges, so the deletion of any edge reduces the crossing n2 K k k number by√ about . Thus, n is -crossing-critical for some that is about cr(Kn) − cr(Kn). This means that there√ are examples of k-crossing-critical graphs G for which cr(G) is about k + k. √ Question 1 Is there a positive constant c for which cr(G) ≤ k + c k whenever G is k-crossing-critical? Equivalently, one may ask if there is√ a constant c such that every graph G has an edge e for which cr(G − e) ≥ k − c k. Recently, Fox and Tóth [18] verified this for dense graphs. More precisely, they proved the following. Theorem 5.3 For any connected graph G with n vertices and m edges, and for e G (G − e) ≥ (G) − m + 1 n + every edge of , cr cr 2 2 1. Fox and Tóth also investigated the effect on crossing numbers caused by the removal of a substantial number of edges in a sufficiently dense graph:

Theorem 5.4 For each ε>0, there is a constant n0, depending on ε, such that, 1+ε if G has n>n0 vertices and m>n edges, then G has a subgraph G with at ( − 1 )m (G) ≥ ( 1 − o( )) (G) most 1 24ε edges for which cr 28 1 cr . Geelen, Richter and Salazar [22] proved that if G is k-crossing-critical, then G has bounded tree-width, thereby directly relating crossing-number problems to graph minors. Crossing numbers are not monotonic with respect to contraction, since contracting an edge might decrease, increase, or not change the crossing number, so this is a significant relationship. (We comment that Robertson and Seymour [46] have found the 41 forbidden minors for those graphs H that are minors of some graph G with cr(G) ≤ 1. However, there are such H for which cr(H ) > 1.) At the same time, Salazar and Thomas made a conjecture that has since been proved by Hlinenýˇ [27]: Theorem 5.5 There is a function f such that if G is k-crossing-critical, then G has path-width at most f(k). We do not wish to get into a discussion of path-width here, but essentially what Theorem 5.5 means is that k-crossing-critical graphs are long and thin. There are several known examples of infinite families of k-crossing-critical graphs, for fixed k, and they are all composed of small pieces ‘stuck’ together in some circular fashion, with more and more copies of the pieces to make the different examples. These have small path-width, which here is essentially the number of vertices in the pieces. These examples suggest a refinement of Theorem 5.5. The bandwidth bw(G) of a graph G is the least integer k for which there is an assignment of 1, 2,...,n 7 Crossing numbers 141 to the vertices such that max |f(v)− f(w)|:vw ∈ E(G) ≤ k.

(Thus, each neighbour of the vertex i must be one of the numbers i − k,i − k + 1, ...,i−1,i+1,i+2,...,i+k.) We make the following conjecture in a formulation by Carsten Thomassen.

Conjecture 1 For each integer k, there is an integer B(k)for which bw(G) ≤ B(k) if G is k-crossing-critical.

Since bw(G) ≤ k implies that G has path-width at most 2k, Conjecture 1 reﬁnes Theorem 5.5. It also implies the following, which itself seems to be challenging.

Conjecture 2 For each integer k, there is an integer D(k) such that the maximum degree of G is at most D(k) if G is k-crossing-critical.

We do not believe that Theorem 5.5 and Conjecture 2 together imply Conjecture 1. Until recently, the following construction gave all known examples of infinite families of crossing-critical graphs. A tile is a triple T = (G,L,R)for which G is a graph and L and R are finite sequences of distinct vertices of G.Atile drawing of such a tile is a drawing of G in the unit square [0, 1] × [0, 1] so that the vertices of L appear on {0}×[0, 1] in order of decreasing y-coordinates and the vertices of R appear on {1}×[0, 1] in order of decreasing y-coordinates. If L and R have the same length, then we can glue two copies (G1,L1,R1) and (G2,L2,R2) of T together by identifying R1 with L2 (in order) to create a new 2 tile T = (G1 ∪ G2,L1,R2). Gluing q copies of T together in this linear way gives the tile T q . The tile crossing number tcr(T ) of a tile T is the minimum value of cr(D) over all tile drawings D of T , and T is planar if tcr(T ) = 0. If T = (G,L,R)is planar, then G is planar, but the converse is not true in general. Let T = (G,L,R)be a planar tile and consider the planar tile T q = (H, L,R). By identifying L with R, but now in the reverse order, we typically create a non- planar graph, denoted by ⊗(T q ). For example, Richter and Thomassen [44] proved that, for the bow-tie B illustrated in Fig. 3, cr(⊗(Bq )) = 3ifq ≥ 3. This can also be proved by a much simpler method, which we now describe. It can be shown that if T is a planar tile, then there is a number N1 such that q if q ≥ N1 and D is a drawing of ⊗(T ) with at most some fixed number c1 of q crossings, then there is another drawing D of ⊗(T ) with c1 crossings for which some copy of T in ⊗(T q ) is drawn as a tile drawing with no crossings. This means that cutting out that copy of T leaves a tile drawing of a tile T˜ q−1 obtained by 142 R. Bruce Richter and G. Salazar

R L1 1

L2 R2

Fig. 3.

gluing together q − 1 tiles, q − 2 of which are T = (G,L,R) and the other is (G,L,R−1) – that is T , but with R in the reverse order. Since cr(⊗T q ) ≤ tcr(T˜ q ) and tcr(T˜ q+1) ≤ tcr(T˜ q ), we easily deduce the following result (see [42]):

Theorem 5.6 If T is a planar tile, then tcr(T˜ q ) = cr(⊗(T q )) for large q.

It is easy to prove that cr(⊗Bq ) = 3. There are four edge-disjoint paths in B from some Li to Ri: from ‘top’ to ‘bottom’ label these P1, P2, P3 and P4. Thus, P1 is disjoint from P3 and P4 and P2 is disjoint from P4. In any tile drawing of ˜ q B , the union of all the P1-paths must cross both the union of all the P3-paths and the union of all the P4-paths, and similarly the P2-paths must cross the P4-paths. This implies that there are at least three crossings. All examples of infinite families of k-crossing-critical graphs known before 2006 can be explained by variations of these arguments; in particular, this is true of the infinite family described in [30]. Oporowski (unpublished) has shown that there is a number N such that, if G is 2-crossing-critical and has more than N vertices, then G is constructed in just the manner of ⊗T q , except that now combinations of thirteen different tiles are used. In recent work, Bokal [10] has introduced the zip product of graphs. Among other things, he has used this [9] to give infinite families of k-crossing-critical graphs, enough such families to show that, for each rational number r ∈ (3, 6), there are infinitely many numbers k for which each member of an infinite family of simple k-crossing-critical graphs has average degree r. Salazar [48] observed that such a rational r must be in the interval (3, 6], so the only open case is r = 6. Richter andThomassen [44] showed that, if the infinite family of k-crossing-critical graphs are all r-regular, then r is 4 or 5. No example of a 5-regular family or a family with average degree 6 is known. Bokal’s families are not made up only of tiles: rather, they decompose into smaller crossing-critical graphs across edge-cuts, which is the operation inverse to the zip product. 7 Crossing numbers 143

6. Other families of graphs

We continue the tile theme. It is clear that, if T is a tile and p and q are positive integers, then p+q p q tcr(T ) ≤ tcr(T ) + tcr(T ). A simple standard result about such sequences, sometimes referred to as Fekete’s lemma, gives a nice limit result: q Theorem 6.1 For each tile T , limq→∞ tcr(T )/q exists. q The average crossing number acr(T ) of a tile T is limq→∞ tcr(T )/q: we note that acr(T ) ≤ tcr(T ). In the next result, from [41], the graph ◦(T q ) is obtained from T q by identifying L and R in the same order. (If T is not planar, then we do not need to reverse R in order to create crossings.) Theorem 6.2 Let T be any tile for which T q exists, for every q. Then there is a constant c = c(T ) such that, for all q,

q q q tcr(T ) ≥ cr(◦(T )) ≥ tcr(T ) − c.

q In particular, limn→∞ cr(◦(T ))/q = acr(T ).

If we knew acr(T ), then it would follow that cr(◦(T q )) is asymptotically equal to n · acr(T ).

Problem Is there any general method for computing (or estimating) acr(T )?

q Let Tp be a tile for which ◦(Tp ) is the Cartesian product Cp Cq of two cycles. It follows from [1] that acr(Tp) = tcr(Tp) = p − 2. Thus, for all q ≥ p, cr(Cp Cq ) lies between (p − 2)q − c(p) and the conjectured value (p − 2)q. An important improvement for this graph has been obtained by Glebski˘ı and Salazar [24]. It is based on [1], but includes some additional arguments:

Theorem 6.3 If q ≥ p(p + 1), then cr(Cp Cq ) = (p − 2)q.

Theorem 6.2 opens up the possibility of getting the right order of magnitude for the crossing numbers of families of graphs. One natural candidate is the generalized Petersen graph P(p,k), obtained from Cp by adding a pendant edge to each vertex and then joining each pair of pendant vertices that are k apart. Theorem 6.2 implies that there is a function f(k)such that cr(P (p, k)) is essentially pf (k). Salazar [49] f(k) ≤ + o(k) f(k) ≥ 1 + o(k) has proved that 2 and 4 . It follows that, roughly, 1 p ≤ (P (p, k)) ≤ p 4 cr 2 . In a different direction, it would be interesting to get some handle on cr(P (q2,q)),asq gets large. It is an easy consequence of Theorem 3.4 and the 144 R. Bruce Richter and G. Salazar main result of [12] that the crossing number of the random cubic graph with n vertices is at least cn2. It would be nice to have actual examples of such graphs. Finally, Bokal [8] has used his zip product to compute the crossing number of the Cartesian product of the star Sp with p leaves and the path Pq with q vertices. His proof is the most succinct argument we know of to show that a particular graph has a given crossing number. An extension of this method to compute the crossing numbers of other Cartesian products of K1,n or wheels with trees or paths is given in [7].

7. Algorithmic questions

In this section, we consider algorithmic aspects of crossing numbers. The basic result in the area is the following, due to Garey and Johnson [21]:

Theorem 7.1 The problem ‘Given a graph G and an integer k,iscr(G) ≤ k?’ is NP-complete.

Hlinenýˇ [28] has recently proved that computing the crossing number of cubic graphs is NP-complete, thereby answering a long-standing question. This proves that the minor-monotone version of the crossing number is also NP-hard (see [11]). The NP-completeness of the crossing number for cubic graphs also follows from a recent result by Pelsmajer, Schaefer and Štefankovicˇ [38], who proved that determining the crossing number of a graph with a given rotation system is NP- complete. A graph is almost planar if there is an edge whose removal leaves a planar graph. It is somewhat surprising that the complexity of determining the crossing number of an almost planar graph is an open question.

Question 2 Is there a polynomial-time algorithm for calculating the crossing number of an almost planar graph?

In this direction, Hlinenýˇ and Salazar have proved that there is a polynomial- time algorithm for approximating (within a constant factor) the crossing number of an almost planar graph with bounded degree. Gutwenger, Mutzel and Weiskircher [26] have shown that if G is planar and if e is an edge not in G, then there is a linear-time algorithm for ﬁnding a planar embedding of G such that the number of crossings that result from the insertion of e is minimized. For ﬁxed k, there is an easy polynomial-time algorithm to determine whether cr(G) ≤ k: try all possible ways of inserting k vertices of degree 4 as the crossings and then test each possibility for planarity. However, a rather different method has been put forward [25], yielding the following result: 7 Crossing numbers 145

Theorem 7.2 For each integer k, there is a linear-time algorithm for determining whether cr(G) ≤ k.

The algorithm begins by reducing G to a graph with bounded tree-width but with the same crossing number, and then test whether the crossing number of G is at most k. (The ﬁrst part works because crossing-critical graphs have bounded tree-width [22].) Closely related is one of the most important algorithmic problems involving crossing numbers, posed by Seese (see the open problems section in [47]).

Question 3 Does there exist a polynomial-time algorithm for computing the crossing numbers of graphs with bounded tree-width?

It is interesting that the problem ‘Is cr(G) ≤ k?’ is trivially in NP, while related problems are not obviously so. For example, consider the rectilinear crossing number rcr(G) of a graph G; this is the minimum number of crossings in any drawing of G in which each edge is a straight-line segment. Bienstock [5] has shown that there is an infinite family Gn of graphs with the property that any straight-line drawing of Gn with rcr(Gn) crossings must have a vertex whose coordinates require more than polynomially many bits. Thus, if we wish to prove that the problem ‘Is rcr(G) ≤ k?’ is in NP, it does not suffice to exhibit a suitable drawing. However, a suitable drawing does suffice to show the problem ‘Is cr(G) ≤ k?’ is in NP. While we are on the subject of rectilinear crossing numbers, Bienstock and Dean [6] have proved the following remarkable results.

Theorem 7.3 (a) If cr(G) ≤ 3, then rcr(G) = cr(G). (b) For each positive integer n ≥ 4, there is a graph Gn such that cr(Gn) = 4 and rcr(Gn) ≥ n.

The pair crossing number problem is also not trivially in NP, since it is not known in advance that if pcr(G) ≤ k, then there is a polynomially describable drawing of G in the plane that demonstrates this. It is now known that there is such a drawing for which each edge has at most 2m crossings [52]. Note that pcr(G) is closely related to the weak realizability problem, which provides as input both G and a set R of pairs of edges, so that crossings may only occur between pairs that are in R. The following is a consequence of [51]:

Theorem 7.4 The problem ‘Given a graph G and an integer k,ispcr(G) ≤ k?’is in NP.

Pach and Tóth [36] have observed that the proof of Theorem 7.1 carries over to prove that ‘odd crossing number’ is also NP-hard. 146 R. Bruce Richter and G. Salazar

8. Drawings in other surfaces

In this section, we consider crossing numbers for drawings of graphs in surfaces other than the plane. Böröczky, Pach and Tóth [13] recently investigated the (plane) crossing number of a graph G embeddable in a surface with Euler characteristic ε. Settling a conjecture of Peter Brass, they proved that G can always be drawn in the plane with at most cεn crossings, where cε is a constant depending on ε and is the maximum degree of G. Examples show that, up to the value of cε, this is best possible. They actually proved a stronger statement: there is a constant cε such d ,d ,...,d G G that, if 1 2 n are the degrees of the vertices of , then can be drawn c n d2 with at most ε i=1 i crossings. This last result is also implied by a recent result of Wood and Telle [62], who proved that every graph with bounded degree that excludes a ﬁxed graph as a minor has linear crossing number. They also proved that every graph with bounded degree and bounded tree-width has linear rectilinear crossing number (cf. Question 3 above). The following was proved by Salazar [50], building upon results of Garcia Moreno and Salazar [20]. The representativity of a graph G embedded in a surface is the minimum number of intersections with G of a non-contractible closed curve γ in , the minimum being taken over all possible curves γ . Theorem 8.1 Let and be surfaces. Then either every -embeddable graph is -embeddable, or there is a constant c such that, if G is a graph embedded in with representativity r, then cr (G) ≥ cr2 + O(r). In the case where is the real projective plane and is the sphere, it should be possible to get a better result. The dual-width of an embedding of a graph in a surface is the length of the shortest non-contractible cycle in the surface dual graph.

Problem Find a good estimate for the (plane) crossing number of a projective planar graph in terms of the dual-width of the embedding.

In [23] it is proved that the desired estimates exist for projective graphs of bounded degree. An algorithmic version of this problem can be given.

Problem Is computing the planar crossing number of projective planar graphs NP-complete?

Part of the motivation for this problem is the fact that the genus of projective planar graphs can be computed in polynomial time (see [17]). 7 Crossing numbers 147

Problem Find a good estimate for the (plane) crossing number of a toroidal graph in terms of the dual-width of the embedding.

Within a constant factor, this has been done for toroidal graphs with bounded degree [29]. One expects that the right estimate for the planar crossing number of a toroidal graph is d1d2, where d1 and d2 are the lengths of transverse non-contractible dual cycles, chosen to minimize the product d1d2. Širánˇ [54] introduced the (orientable and non-orientable) crossing sequence of a graph G. Recall that the orientable surface of genus h is denoted by Sh and the non-orientable surface of genus k is denoted by Nk. Then the (G), (G), (G),... orientable crossing sequence is crS0 crS1 crS2 and the non- (G), (G), (G),... orientable crossing sequence is crS0 crN1 crN2 . Evidently, for h ≥ γ G G (G) < (G) ( ) (the genus of ), crSh crSh+1 , and similar results hold for the crosscap number. The main result of [54] is that if the (ﬁnite) sequence a0 >a1 >a2 > ··· > ak = 0 of integers is convex (that is, ai+1 − ai+2 ≤ ai − ai+1 for each i), then there is a graph G whose orientable crossing sequence is (a0,a1,a2,...,ak, 0, 0, 0,...). There is also a graph H whose non-orientable crossing sequence is (a0,a1,a2,...,ak, 0, 0, 0,...). Archdeacon, Bonnington and Širánˇ [3] showed by example that not every crossing function is convex. Furthermore, they proved that if a0 >a1 > 0, then there is a graph G that embeds in the Klein bottle satisfying crNi (G) = ai, for i = 0, 1. In the orientable case, DeVos, Mohar and Šámal [14] have proved that if a0 >a1 > 0, then there is a graph G that embeds in the double torus and crSi (G) = ai, for i = 0, 1. Archdeacon et al. [3] suggested the following conjecture: Conjecture 3 Any strictly decreasing sequence of positive integers is the orientable crossing sequence of some graph and the non-orientable crossing sequence of some graph.

9. Conclusion

We have attempted to highlight recent work about crossing numbers that demonstrate the growth of the theory. We have seen: • relations with extremal graph theory; • Székely’s simple proof of the Erdos–Trotter˝ theorem, which connects crossing numbers with several areas of mathematics; • our understanding of k-crossing-critical graphs grow substantially; 148 R. Bruce Richter and G. Salazar

• intimations of how to get good estimates for the crossing numbers of the members of families of graphs; • many advances in algorithmic and complexity questions.

Despite these exciting results, there are still many easily stated problems to motivate further work.

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TOMAŽ PISANSKI and ARJANA ŽITNIK

1. Introduction 2. Representations of graphs 3. Energy and optimal representations 4. Representations of maps 5. Representations of maps in the plane 6. Representations of incidence geometries and related topics References

Motivated by graph drawing problems, we can formulate a theory of graph representations precisely, admitting various generalizations. Each vertex is represented as a point in Euclidean space and each edge as a line segment spanned by the representations of its endpoints. A typical problem in this area is ﬁnding an optimal graph representation that minimizes a given energy function. For some parameters this problem admits an exact solution while other parameters require iterative methods. Representations of graphs are then extended to representations of maps and other incidence geometries.

1. Introduction

One reason why the pure combinatorial structure of a graph is so interesting and attracts the attention of many researchers lies in the fact that a graph can usually be completely deﬁned geometrically. On the other hand, some physical and geometrical objects, such as molecules and polyhedra, carry a graph structure. Usually a graph is depicted by placing its vertices in the plane (or in 3-space) and joining the points corresponding to adjacent vertices of the graph with line segments or simple curves. Based on this idea we make the notion of graph representation precise.

151 152 Tomaž Pisanski and Arjana Žitnik

The notion of graph representation was first used implicitly in the work of Steinitz [45], Wagner [49], Fáry [14] and Stein [43]. They produced the following fundamental results in the theory of simple planar graphs. Theorem 1.1 (Steinitz’ theorem) A graph is planar and 3-connected if and only if it is the graph of a convex 3-dimensional polyhedron. Theorem 1.2 (Fáry’s theorem) Every planar graph can be represented in the plane without edge crossings with line segments for edges. However, the idea of graph representation was first explicitly considered by Tutte [47], [48] for planar 3-connected graphs, where there was no need to distinguish between representations of graphs and representations of maps: namely, each 3-connected planar graph uniquely determines its plane map up to the choice of the outer face. Parsons and Pisanski [36] were the first to consider the notion of vector representation of graphs in a quite general setting with no explicit reference to maps. We introduce representations of graphs in Section 2. In Section 3 the concept of the energy of graph representation is defined. This concept is crucial in graph drawing since representations minimizing energy with given restrictions are usually the ones produced by certain graph-drawing algorithms. In Section 4 we generalize the idea of representation of planar maps to maps on other surfaces, while Section 5 deals with planar representations of such maps. We conclude the chapter with a brief discussion of representations of vertex- coloured graphs (alias incidence geometries), which may be regarded as a common generalization to graphs and maps with non-trivial application to configurations. If we apply this new concept to graphs as incidence structures, the resulting definition of a graph representation is much more general. In particular, it allows us to remedy some of the drawbacks of the initial concept of graph representation, which is rather restrictive since it does not allow us to describe more sophisticated drawings of graphs, such as those with curved lines that arise naturally, for instance, in drawings of maps on surfaces. The aim of this chapter is to motivate interested researchers to develop a representation theory of topological and geometrical graph theory, focused on some aspects of graph drawing. More on the subject of graph drawing can be found in the book by Di Battista et al. [12] and that edited by Kaufmann and Wagner [28].

2. Representations of graphs

We start by deﬁning graph representation in an arbitrary Euclidean space. Let Rr be Euclidean r-dimensional space, with d denoting the usual distance between points, 8 Representing graphs and maps 153

d( , ) =|| − || = r (a − b )2. a b a b i=1 i i Let G = (V, E) be a graph and let P(X) denote the power set of a given set r r X. The pair of mappings (ρV ,ρE), ρV : V (G) → R , ρE : E(G) → P(R ),is called a graph representation or an Rr -representation of a graph G if, for any edge e = vw ∈ E(G), its representation ρE(e) is the line segment joining ρV (v) and ρV (w); in other words:

ρE(e) ={λρV (v) + µρV (w): λ, µ ≥ 0,λ+ µ = 1}.

We deﬁne the length ||e|| of the edge e = vw relative to ρ as the length of the corresponding line segment, or equivalently as ||e|| = d(ρ(v), ρ(w)). Since the edge-representation is completely determined by the vertex-representation, we usually drop the subscripts and denote both mappings simply by ρ. An R2-representation is called planar and an R3-representation is called spatial. A planar representation is therefore equivalent to the usual drawing of a graph in the plane with straight edges. A representation is non-degenerate if the mapping ρV is one-to-one. If we want to reconstruct a graph from its representation (drawing), the mapping ρV must be one-to-one; hence the representation has to be non-degenerate. However, there exist ambiguous non-degenerate planar representations, as in Fig. 1(c), where the image does not determine the graph C5 uniquely. A representation is strong if, for every edge e and vertex v, ρ(v) lies in ρ(e) if and only if v is an endpoint of e. A representation is an embedding if the representations of two edges have a common point only if the edges are incident with a common vertex. In particular, a planar representation that is an embedding into the plane is called a plane representation. Clearly only planar graphs admit plane representations. However, non-planar graphs also admit planar representations. Fig. 1 shows various representations of C5.

4 4 5 3 5 3 5 3 5

1 2 1 2 4 3 4 1 2 1 2 (a) (b) (c) (d)

Fig. 1. Representations of C5: (a) plane, (b) strong, (c) non-degenerate, (d) degenerate 154 Tomaž Pisanski and Arjana Žitnik

An interesting class of representations constitute representations that have all the edges of the same length. They are called unit-distance representations or unit-distance graphs. (Sometimes the term ‘unit-distance graph’ requires that two vertices be adjacent if and only if their distance apart is 1.) For instance, such representations naturally arise in chemistry, where all bonds are expected to be of the same size.

Example 1 Fig. 2 depicts two planar representations of the generalized Petersen graph G(6, 2), known also as the ‘Dürer graph’. The drawing on the right-hand side is a unit-distance representation.

Fig. 2. The generalized Petersen graph G(6, 2) and its unit-distance representation

In a certain sense, this generalizes to an arbitrary simple graph G on n vertices. Since G is a subgraph of the complete graph Kn, it suffices to find a unit-distance representation for Kn. For any r ≥ n − 1, the complete graph Kn admits a unit- r distance representation in R . However, K4 already has no unit-distance planar representation:√ the ratio between the longest and the shortest edge is always at least 2, which is attained if the four vertices of K4 are placed at the four corners of the unit square. Motivated by the above example we introduce the following concept: in the case when ||e|| > 0 for every e ∈ E(G), we define the dilation coefficient D(ρ) as the ratio of the longest edge to the shortest edge. Note that D(ρ) ≥ 1. In the special case when D(ρ) = 1, this representation is a unit-distance representation. A natural question arises: given a graph G, how can we find a planar non- degenerate representation minimizing the dilation coefficient? Define D(G) to be the (planar) dilation coefficient of G. It is obtained as the minimum of D(ρ)√ over all non-degenerate planar representations ρ of G, for example, D(K4) = 2. Clearly, we may restrict our search to representations whose shortest edge- length is 1. This can be described in a more formal way. Two representations ρ and ρ of G are equivalent if there exists a constant k>0 such that d(ρ(v), ρ(w)) = kd(ρ(v), ρ(w)), for any pair of vertices v and w. Obviously, two equivalent representations have the same dilation coefficient. 8 Representing graphs and maps 155

Several graph-drawing algorithms determine a representation of a graph G by an optimization process, as described in the next section. Each representation obtained in this way gives an upper bound on D(G); however, lower bounds are much harder to obtain.

3. Energy and optimal representations

Graph representations provide an interesting source of discrete optimization problems. Usually the goal is to find, among the representations of given type, one that is optimal in a certain sense. To this end we define a real-valued ‘criterion function’ E(ρ) that can be used for comparing the representations. When the minimum is sought, such a function is called an energy function. To prevent the representation collapsing to a single point, we have to choose a convenient normalization as a subsidiary condition. Different choices of energy function and normalization usually lead to different optimal representations of the same graph. In this section we show that representations of graphs and energy models constitute a suitable environment for studying various drawing algorithms as energy minimization problems. Let ρ be a representation of a graph G in Rr and p>0, and define the energy function Ep(ρ) as /p p 1 Ep(ρ) = ||e|| . e∈E(G) In the limit when p →∞,weget

E∞(ρ) = max ||e||. e∈E(G)

Aconvenient normalization in this case is to put mine∈E(G) ||e|| = 1. Then D(ρ) = E∞(ρ). Godsil and Royle [17] deﬁned the energy of a representation ρ equivalently to 2 our E2(ρ) and we denote this energy by EE(ρ). For a weighted graph in which each edge e has a positive weight ωe, the energy of a representation is deﬁned as 2 EE(ρ, ω) = ωe||e|| . (1) e∈E(G)

In a vector space, such as Rr , we can regard the vectors ρ(v)for a graph G of order n as row vectors, and we represent ρ by the n × r matrix P with the images of the r vertices of G as the rows. A representation ρ in R is orthonormal if PTP = Ir . n ≥ r. ρ Clearly, for an orthonormal representation, A representation is balanced if v∈V (G) ρ(v) = 0. In other words, a representation is balanced if and only if 156 Tomaž Pisanski and Arjana Žitnik its barycentre is the origin. Balanced orthonormal representations are considered in the next subsection.

The Laplace method Let G be a connected weighted graph on n vertices with a weight function ω that assigns a positive real number to each edge. The n × n matrix Q with elements  −ω vw ∈ E(G),  vw if q = − q v = w, vw  uv∈E(G) uv if 0 otherwise is called the generalized Laplacian of a weighted graph G. In the case when all the weights are equal to 1 (that is, the graph is not weighted), the matrix Q is called its Laplacian. Let ρ be a representation of G in R. Then ρ is a column vector in Rn, and we observe that Q is the matrix for the positive semi-deﬁnite quadratic form EE(ρ). Since the row-sums of Q are all zero, Q has 0 as an eigenvalue, but all other eigenvalues are positive. Moreover, if ρ is a unit vector and balanced (so ρ(v) = ρ(w) for some v, w), then by the principal axis theorem [42], the smallest possible value for EE is the smallest positive eigenvalue. In general, we have the following result (see [39], [17]):

Theorem 3.1 Let G be a connected weighted graph with generalized Laplacian Q, where the eigenvalues of Q are λ1 ≤ λ2 ≤ ··· ≤ λn and λ2 > 0. Then r the minimum energy EE of a balanced orthonormal representation of G in R is λ2 + λ3 +···+λr+1.

The orthonormal representation ρ of Theorem 3.1 is given by the matrix [x2, x3,...,xr+1], whose columns are composed of orthonormal eigenvectors corresponding to λ2,λ3,...,λr+1. For r = 2 or 3, this gives a representation in 2-space or 3-space. Examples of such representations are shown in Fig. 3, where an R2-representation of the dodecahedron is obtained by using the second and third eigenvectors of the Laplacian, and an R3-representation of the graph F180 is obtained by using the second, third and fourth eigenvectors. The graph F180 on 180 vertices is an example of a fullerene, a cubic planar graph whose faces are only pentagons and hexagons. Its construction is described in Section 4. Any procedure that obtains a representation of a graph as a solution of the eigenvector problem is called the eigenvector method. In particular, the eigenvector method applied to the Laplacian, choosing only the second and third and possibly the fourth eigenvectors, is called the Laplace method. Theorem 3.1 guarantees that 8 Representing graphs and maps 157

(a) (b)

Fig. 3. (a) An R2-representation of the dodecahedron, and (b) an R3-representation of the fullerene F180, both obtained by the Laplace method the Laplace method produces a representation that minimizes the energy given by (1). The Laplace method does not always produce the best results. One reason for this lies in eigenvalue degeneracies, where orthogonal eigenvectors are not uniquely determined. That is why it makes sense to study the combinatorial structure of the eigenspaces. One way to do this is with nodal domains (see [16]).

1-dimensional representations and nodal domains In a 1-dimensional representation of a graph G each vertex is assigned a real number. These numbers can be collected in a vector x with entries labelled by the vertices of G. The positive support supp+(x) consists of those vertices v for which xv > 0, and the negative support supp−(x) consists of those vertices v for which xv < 0. Furthermore, the weak positive support wsupp+(x) consists of those vertices v for which xv ≥ 0, and the weak negative support wsupp−(x) consists of those vertices v for which xv ≤ 0. A strong nodal domain for x is a connected component of the graph induced by a set of strong support vertices, and a weak nodal domain is similarly deﬁned. We can visualize such a representation by drawing vertices as discs in two colours, the colour of the disc corresponding to a vertex v depending on the sign of xv, and the radius of the disc being proportional to |xv|. From such a drawing it is possible to read off the strong and weak nodal domains.

Example 2 Let us consider the generalized√ Petersen graph G(10, 4). The second smallest eigenvalue of its Laplacian 3− 5 has multiplicity 1 and its third smallest √ eigenvalue 3 − (5 + 5)/2 has multiplicity 2. Fig. 4 depicts its R3-drawing according to the Laplace method and the nodal domains along each of the three eigenvectors used in the drawing. Since the multiplicity of the third smallest eigenvalue is 2, the corresponding nodal domains are not uniquely determined. 158 Tomaž Pisanski and Arjana Žitnik

Fig. 4. An R3-representation of the generalized Petersen graph G(10, 4) and the corresponding 1-dimensional representations with nodal domains

It seems that nodal domains play an important role in best eigenvector selections. Eigenvectors with fewer nodal domains usually give better pictures. For higher-dimensional representations the nodal domains of each coordinate projection must ﬁt together appropriately, in order to produce a pleasing drawing. For molecular toroidal graphs, experimental results may be found in [20]. The most powerful result in this area is the following theorem of Gladwell et al. [16], which represents a discrete analogue of the well-known Courant nodal domain theorem for differentiable surfaces.

Theorem 3.2 (Graph nodal domain theorem) Let G be a graph with λ1 <λ2 < ··· <λr being the eigenvalues of its Laplacian. Assume λk (1

Both bounds in the above theorem are sharp. The theorem gives another justification as to why first eigenvectors are a good choice for a representation. On the other hand, special care must be taken in the case of degenerate eigenvalues. The nodal domain structure of eigenspaces of graphs can be best understood via hyperplane arrangements or oriented matroids (see [50]). Some specific results for hypercubes can be found in [4].

The Tutte method

A cycle C of a graph G is peripheral if it is induced and G − C is connected. For example, any face of a 3-connected planar graph can be shown to be a peripheral cycle. We say that a representation ρ of a graph G is barycentric relative to a set of vertices S if, for each v/∈ S, the vector ρ(v) is the barycentre of the images of the neighbours of v. The following lemma and theorem are adapted from Tutte [48]. 8 Representing graphs and maps 159

Lemma 3.3 Let G be a connected graph, let S be a non-empty set of vertices of G, and let σ be a map from S into Rr . Then there is a unique r-dimensional representation ρ of G that extends σ and is barycentric relative to S. Moreover,this representation has minimum energy EE among all r-dimensional representations of G that extend σ . Note that the system of equations that ρ must satisfy to be barycentric has r ·|V (G) \ S| equations and variables: ρ(v) = 1 ρ(w), v ∈ V (G) \ S. (v) (2) deg w∈N(v)

Also, since the case in which S is mapped to a single point a ∈ Rr clearly has the unique solution ρ(v) = a for all the vertices v when G is connected, the matrix for the system is non-singular. Theorem 3.4 Let G be a 3-connected graph and let C be a peripheral cycle in G with k vertices. Let σ be a mapping from V(C) to the vertices of a convex k-gon in R2 so that adjacent vertices in C are adjacent in the polygon. The unique barycentric representation that extends σ determines a representation of G in R2. This representation is plane if and only if G is planar. A barycentric drawing based on this theorem is obtained by solving the system of equations (2). The drawings given by Theorem 3.4 are sometimes called Tutte drawings of a graph (see [17]). They are related to the notion of the Schlegel diagram of a polyhedron – that is, a graph obtained by projecting the graph of the polyhedron to one of its faces from a point very near the face but outside the polyhedron. Every Schlegel diagram is a weighted barycentric Tutte embedding (with convex outer face) and vice versa. The main problem with Tutte drawings is that vertices tend to crowd in the centre. In order to avoid this, we introduce appropriate weights on the edges, which depend on the distances from a given set of vertices. Let S be a set of vertices of G and let σ be a map from S into Rr . For each vertex v in G, let δ(v) be its distance from S – that is, the length of a shortest path from v to a vertex of S. On the edges that do not have both endpoints in S, deﬁne ωvw = φ(δ(v), δ(w)), for a suitable positive symmetric function φ ω = +|δ(v) − δ(w)|p ω = {δ(v), δ(w)}−p (such as vw 1 or vw max , for some real number p). Let ωvv = vw∈E(G) ωvw for v ∈ S. The corresponding weighted barycentric representation ρ extending σ is called the Schlegel representation of G with respect to S. It is deﬁned by the system of equations

1 ρ(v) = ωvwρ(w), for v ∈ S. ωvv w∈N(v) 160 Tomaž Pisanski and Arjana Žitnik

(a) (b)

2 Fig. 5. Two R -representations of the fullerene graph F180

Fig. 5 shows a comparison between the Tutte and Schlegel drawings of the fullerene graph F180; the second drawing was obtained by the Schlegel method with ωuv = max{δ(u), δ(v)}−5/2.An iterative algorithm for drawing Schlegel diagrams, based on similar ideas, was developed by Plestenjak [40].

Iterative methods A major challenge facing the ‘graph-drawing community’ is to devise algorithms for drawing graphs that arise from ‘real-life’ applications. Such graphs are usually large and lack internal structure. Even those that possess a structure (such as graphs of printed boards or genealogical trees) often require special considerations: interested readers are referred to DiBattista et al. [12] for extensive information on this subject. Several software packages for drawing different types of graphs are described in Jünger and Mutzel [26]: one such example is the program Pajek (see [3], [26]), which has been used to produce several of the prize-winning graph drawings that accompany annual Graph Drawing conferences; see, for instance, [23] and [6]. Among the most practical devices are the so-called spring embedders, also known as a force-directed placement. The main idea is to consider edges as springs, define two forces fr and fa, one pushing vertices apart and the other attracting adjacent vertices; the forces between two vertices depend on the Euclidean distance between them. An energy function is defined that takes into account the contributions of all pairs of vertices.Aminimum of the energy function corresponds to an equilibrium point of the forces. The problem of energy minimization is then solved by local optimization or simulated annealing techniques. This method has many variations and adaptations; for instance, the positions of some vertices may be fixed or, instead of the plane, 3-dimensional (or even higher-dimensional) space may be used. Each of these methods defines a graph representation. 8 Representing graphs and maps 161

The approach of Kamada and Kawai [27] gives an explicit energy function. If d(v,w) denotes the graph distance between the vertices v and w in a graph G, then the energy of a representation ρ is given by

||ρ(v) − ρ(w)|| − d(v,w) 2 E(ρ) = . d(v,w) v,w∈V

Sometimes it is hard to separate the optimization problem from the method for its solution. This may happen when the graph-drawing algorithm is given without the energy function being stated explicitly. For instance, Fruchterman and Reingold [15] spoke of attractive and repulsive forces, but never explicitly stated their energy function. Let k be a constant representing the ‘ideal’ length of all the 2 edges and let =||ρ(v) − ρ(w)||. Then the attractive force is fa() = /k and 2 the repulsive force is fr() = k /: fr acts on any pair of vertices, while fa acts only between two adjacent vertices. The following formula is a possible energy function for their algorithm:

||ρ(w) − ρ(v)||3 E(ρ) = − k2 ||ρ(w) − ρ(v)|| . k log v∈V w∈N(v) 3 w∈V

Davidson and Harel [10] used a similar (but more general) approach with uncoupled constants and additional terms for the energy that penalizes the number of edge-crossings, the closeness of a vertex to an edge and the closeness of vertices to the boundary of the ‘canvas’ on which the drawing takes place. The equilibrium was found by the method of simulated annealing. Eades [13] introduced the idea of a force-directed placement method with the functions 2 fa() = k1 log(/k2) and fr() = k3/ , where k1,k2 and k3 are suitable constants. Plestenjak [40] fixed the vertices of a given cycle and defined fa() = 3 k between pairs of adjacent vertices and fr() = 0. However, for every pair of adjacent vertices, the coefficient k depends on their graph-theoretical distance to the cycle, in order to prevent congestion of vertices towards the centre of the cycle. In molecular mechanics, a goal is to find coordinates for molecular graphs and ideas similar to those of the graph-drawing community are used. The main difference is that graphs are usually represented in R3 rather than in R2 and knowledge from chemistry, such as angles between incident edges, is used. The most common form of the energy is 162 Tomaž Pisanski and Arjana Žitnik

2 2 E(ρ) = kb(r − r0) + kθ (θ − θ0) + A(1 + cos(nτ − ϕ)) edges angles torsions

−A B q q + i,j + i,j + i j . r6 r12 r i j i,j i,j i j i,j

The ﬁrst term represents stretching energy, the second bending energy, the third torsion energy, and the last two terms non-bonded interaction energy, composed of the van der Waals term and the electrostatic term. Constants are determined by comparison with the results obtained by more accurate ab initio quantum- mechanical methods. Delgado Friedrichs [11] presented sophisticated heuristics used in the program CaGe for fast realizations of certain connected planar graphs such as fullerenes in two or three dimensions. He combined several clever ideas into a procedure that works in phases, each of which minimizes a different energy function. The method uses plane embeddings of graphs, and this makes it the ﬁrst method in this class to use properties of faces to determine the representation. Here are some of the best tricks:

• instead of calculating the repulsive force between all pairs of vertices, which takes O(n2) time, only the force between vertices and the barycentre of the graph is needed, and this takes only linear time O(n); • vertices at distance 2 in the graph may be used to account for the angles between adjacent edges – this idea is also used in a different setting in [19]; • plane graphs (actually, maps) can be triangulated in order to achieve 3- connectivity; • the most innovative idea is to use areas of auxiliary triangles in the energy calculation, thereby avoiding the drawback of the Tutte method, which makes some faces much smaller than others.

The energy functions from previous subsections admit closed form optimization solutions. The energy models here usually require iterative heuristic algorithms that find near-optimal solutions. We conclude this section with a generic algorithm for finding optimal graph representations that can be adapted and fine-tuned for a chosen energy function model. This algorithm can be readily adapted to specific energy functions or constraints in a way that may considerably improve the speed; the reader is referred to the original articles, where further details of implementation can be found. A random choice of δ(v) ∈ Rr and λ(v) ∈ R will give poor results in terms of overall time complexity, while seemingly more intelligent choices may reduce the number of iterations but may slow down each step of computation. 8 Representing graphs and maps 163

Generic iterative graph representation algorithm with cooling schedule

Input: graph G, dimension of representation r, energy function E, set S of ﬁxed vertices of G, prescribed coordinate vectors σ for vertices in S. Parameters: maximum displacement D, number of iterations N, temperature cooling schedule T(0) = 1 ≥ T(1) ≥ ... ≥ T(N)>0. r Output: representation ρ of G in R with small, near-optimal energy E(ρ). Pseudocode: 1. For each vertex v ∈ S, let ρ(v) = ρ (v) := σ(v). r 2. For each vertex v ∈ S choose a vector ρ(v) ∈ R . 3. For k = 1, 2,... ,N,do a. For each vertex v ∈ S, repeat r (i) choose some vector δ(v) ∈ R ; (ii) choose some number λ(v) ∈ R such that |λ(v)|≤D · T(k); (iii) let ρ (v) := ρ(v) + λ(v)δ(v). b. If E(ρ )

For S = ∅, one gets the usual drawings, such as Eades, Kamada–Kawai, Fruchterman–Reingold or Davidson–Harel. For S = ∅, one gets various Tutte representations or Schlegel diagram drawings. The algorithm can be implemented in such a way that at each step a new drawing of the graph is shown on the screen. Anyone trying to implement such an algorithm should select a suitable projection from Rr to the screen and should be warned about the possibility of the graph drifting away, collapsing to a single point (for instance if S = ∅ and no repulsive forces are present), or expanding to inﬁnity if the graph is not connected.

4. Representations of maps Flags and involutions A map is a cellular embedding of a graph in a surface, with the emphasis on the incidence structure of vertices, edges and faces. The surface is divided into ‘right’ triangles by joining the centre of each face to the midpoint of each incident edge and to each incident vertex as in Fig. 6. The resulting triangles are called flags. Abstractly, a map M is defined by a set of flags (M) and three fixed-point-free involutions τ0,τ1 and τ2 on (M) with the following properties:

• τ0τ2 = τ2τ0 and this is ﬁxed-point-free; • the group τ0,τ1,τ2 acts transitively on . 164 Tomaž Pisanski and Arjana Žitnik

τ w eЈ 1( ) w

v e vЈ τ2(w)

fЈ

Fig. 6. A ﬂag ϕ may be viewed as a triangle with vertices vϕ,eϕ,fϕ; the ﬂags τ2(ϕ), τ1(ϕ), τ0(ϕ) are represented by the three triangles adjacent to ϕ

Involutions can be viewed as instructions for gluing together the flags to make the map. The orbits V(M)of τ1,τ2 are called vertices, the orbits E(M) of τ0,τ2 are called edges, and the orbits F(M) of τ0,τ1 are called faces (compare, for instance [17]). Given a flag ϕ,weletvϕ,eϕ and fϕ be its vertex, edge and face orbits. The pair (V (M), E(M)) determines a graph G = skel(M), the 1-skeleton of the map, such that a vertex orbit v is incident with an edge orbit e if and only if they share a flag. We would like to identify each flag ϕ uniquely by the triple (vϕ,eϕ,fϕ), but if there are loops in G or its dual, this cannot be done. For example, Fig. 7(a) shows a map for which the 1-skeleton (thick lines) has a loop; consequently, two flags have the same vertices v, e and f . Similarly, Fig. 7(b) shows a map for which the 1-skeleton of the dual (thin lines) has a loop and two flags have the same vertices. If there are no loops in G or its dual, we call the map M flag-simple. The reason for introducing flag-simple maps is that only they can be regarded as incidence geometries (see Section 6).

v w e e v w wЈ wЈ f

Fig. 7. Maps that are not ﬂag-simple 8 Representing graphs and maps 165

If M is ﬂag-simple and we identify ϕ by a triple (v,e,f), then

τ0(v,e,f)= (v ,e,f), τ1(v,e,f)= (v, e , f ), τ2(v,e,f)= (v,e,f ), for appropriate v ∈ V,e ∈ E,f ∈ F (see Fig. 6). On the union V ∪E∪F we can define an incidence relation as follows: two different elements are incident if and only if they belong to some triple ϕ ∈ . This irreflexive and symmetric incidence relation defines on the vertex-set V ∪ E ∪ F a simple graph that we call the Levi graph L(M) of the map M. It is in fact a triangulation of the surface obtained by a barycentric subdivision of the map, 3-coloured by the symbols V,E,F. There is a dual viewpoint that works for flag-simple maps. In the Levi graph of a map M the flags are represented as triangles, while the vertices fall in three sets: V(M), E(M) and F(M). In the connected cubic flag graph F(M), the vertices are flags and the edges are coloured with the three colours τ0,τ1,τ2, so that two edges incident with the same vertex get different colours. Moreover, the subgraph of F(M) determined by the edges coloured with τ0 and τ2 consists of disjoint 4-cycles. The edges of each colour class form a 1- factor, which can be viewed as a fixed-point-free involution. Hence, we may view τ0,τ1,τ2: (M) → (M), as three fixed-point-free involutions with the extra condition that τ0τ2 is a fixed-point-free involution, thereby returning to our original definition of the map.

Representations of maps from graph representations In fact, the pioneering work of Tutte on planar graph representations in R2 deals with representations of planar maps. He considered the so-called convex representations, in which the edges are represented as line segments and the faces are convex polygons in the plane, apart from the unbounded face whose complement is a convex set. The easiest way to introduce representations to maps is via graphs. A representation of a map M is simply a representation of its 1-skeleton skel(M). However, such a representation depends only on the vertices and edges and not on the faces. If we want to use faces and still stay within the framework of graphs, we can define the representation of M as a representation of either its Levi graph or its flag graph. In such a case, we can define the energy as explained in Section 3. Another possibility is to start with the representation of the 1-skeleton G = skel(M), and then treat faces separately, as new types of objects. In this case we r extend the pair of mappings (ρV ,ρE) with ρV : V (G) → R and ρE: E(G) → r r P(R ) to the triple (ρV ,ρE,ρF ), where ρF : F(M) → P(R ) and for each edge e incident with a face f we require that ρE(e) ⊆ ρF (f ). We require that each edge be represented by the convex closure of its endpoints and each face by the convex closure of its vertices; such a representation is sometimes called a convex 166 Tomaž Pisanski and Arjana Žitnik representation. In addition, we usually assume that each face is represented by a plane polygon and that the intersection of two polygons is at most 1-dimensional. However, we are faced with the problem of how to define the energy of a map representation that depends not only on vertices and edges but also on faces. In general, it can be defined in such a way that the contribution of a non-planar face is much higher than that of a planar face. This can yield efficient heuristic algorithms for finding convex representations with planar faces (see [38]). The requirement that the faces be represented as planar polygons is being used by geometers, for instance in the study of uniform polyhedra (see [30]). In the problem of polyhedral realizations of maps, it is required that the faces be convex and non-intersecting (see [5]). If face intersection is allowed, then (usually) no edge is allowed to lie in the interior of a face. A convex representation of M in the plane R2 implies that M is planar. In this sense, our definition of a convex representation coincides with the one given by Tutte. However, there are non-planar maps that admit convex representations in R3, as the following example shows.

Example 3 The projective-planar uniform polyhedron tetrahemihexahedron is shown in Fig. 8; its 1-skeleton is the octahedron K2,2,2. The map is composed of three pairwise intersecting quadrilaterals, ABCD, AECF and BFDE, that pass through the centre of the polyhedron, and four triangles AEB, AFD, BCF and CDE, only two of which are visible in the ﬁgure.

C A B

Fig. 8. A tetrahemihexahedron

In the sequel we brieﬂy explain the energy model of the CaGe-method that uses areas of faces (see [11]). Instead of graphs, maps are used. In particular, we need the notion of the 2-dimensional subdivision of a map. A 2-dimensional subdivision Su2 is obtained from a map by adding a vertex in the barycentre of each face and connecting it to all vertices of that face. For any real representation ρ of M,we 8 Representing graphs and maps 167 can deﬁne the energy E(ρ) (paraphrasing the CaGe-method) as (A(ρ, Q)||c(ρ, Q) − ρ(v)||)2 E(ρ) = Q∈F(M ),v∈Q , A(ρ, Q)2 v∈V(M) Q∈F(M ),v∈Q where M is a 2-dimensional subdivision of M, M = Su2(M), and Q ={u, v, w} is a triangular face in F(M) whose represented triangle ρ(Q) has area A(ρ, Q) c(ρ, Q) = 1 (ρ(u) + ρ(v) + ρ(w)) and barycentre 3 . Delgado Friedrichs did not explicitly specify the energy and used it only for 2-dimensional representations of planar maps, with the area calculated using vector products

A(ρ, Q) = 1 ||(ρ(v) − ρ(u)) × (ρ(w) − ρ(u))||. 2

However, there is no reason why it should not be used in higher dimensions.

Operations on maps Using the formal definition of a map in terms of flags, we can now define several transformations or operations on maps. The most important among these are the dual, the truncation, the 1-dimensional subdivision and the medial.

The medial map Intuitively, given a map M realized as a polyhedron, think of cutting away each vertex v by a plane v in such a way that for each edge e = vw the corresponding planes v and w intersect at the midpoint of e. For maps of maximum degree greater than 3 this construction may be geometrically impossible, since each plane is determined by three points, but the topological or combinatorial construction is well defined. The resulting map is 4-regular, resides in the same surface as M, and has two types of faces: those corresponding to the original faces and those corresponding to the original vertices; in particular, a vertex v of degree k gives rise to a k-gonal face. For example, the medial of a tetrahedron is an octahedron, and the medial of an octahedron is a cuboctahedron. We define the medial operation Me formally, using flags. Let (v,e,f)be the triple of vertices of a triangle, corresponding to a flag ϕ. We subdivide this triangle into two triangles, (v,e,f) corresponding to ϕ, and (v,e,f) corresponding to ϕ, where v = v = e, e = e, f = v and f = f (see Fig. 9). The new vertex e = e is located on the edge vf of the original triangle. If it is placed in the barycentre, we can write it as e = e = (v + f)/2. We treat the entries v, e and f as formal variables that can be added, and this gives a shorter description of this subdivision: 168 Tomaž Pisanski and Arjana Žitnik

f f Љ

eЈ= eЉ

v e f' vЈ= vЉ

Fig. 9. The medial operation Me is deﬁned as a subdivision of ﬂag triangles

            v 010 v v 010 v e  =  1 1  e  e  =  1 1  e  . 2 0 2 and 2 0 2 f 100 f f 001 f

The medial operation can be described by the matrices     010 010 =  1 1  =  1 1  . A 2 0 2 and A 2 0 2 (3) 100 001

Operations and their matrices In a similar way, we can deﬁne other operations on maps, including the dual and truncation. The following table gives deﬁnitions of some well-known operations. operation symbol example matrices   100 identity Id 010 001     010 010  1 1  ,  1 1  medial Me 2 0 2 2 0 2 100 001   001 dual Du 010 100       1 1 0 1 1 0 1 1 2 2 2 2 2 2 0 truncation Tr  1 1  ,  1 1  ,   2 0 2 2 0 2 010 100 001 001     100 010 1-dimensional Su1  1 1 0 ,  1 1 0 subdivision 2 2 2 2 001 001 8 Representing graphs and maps 169

Combinations of operations If we combine some of these transformations, some other interesting transformations result. In chemistry, the operation Le = Tr ◦ Du is sometimes called the leapfrog transformation. For instance Le (dodecahedron) is the well-known fullerene F60, a model for the pure carbon molecule discovered in the 1980s. The graph F180 in Figs. 3 and 5 is Le(F60). The operation Du◦Me was studied by Mohar and Rosenstiehl in [34] under the name of angle transformation. An interesting operation is the 2-dimensional subdivision Su2 = Du ◦ Tr ◦ Du. The well-known barycentric subdivision plays an important role in mathematics and can be defined as BS = Su2 ◦ Su1; given a map M, the dual of BS(M) is the flag graph of M. The following result is useful for the automatic determination of the set of matrices for combined operations. Theorem 4.1 Let S and T be the operations defined by the sets of matrices {A1, A2,...,Ar } and {B1, B2,...,Bs}, respectively. Then the combined operation S ◦ T is defined by the set of rs product matrices {AiBj : i = 1, 2,...,r; j = 1, 2,...,s}. Using this result one can easily show that, for any map M,Du(Du(M)) = M and Me(M) = Me(Du(M)). Note that the choice of matrices is not unique, and that seemingly minor changes in the definition may result in operations having different properties. For instance, instead of taking A and A as in (3) in the definition of Me, we could have taken     010 010 =  1 2  =  1 2  ; B 3 0 3 and B 3 0 3 100 001 in this case the result Me(M) = Me(Du(M)) no longer holds.

Operations and representations The matrices that we used for deﬁning operations on maps can also be used to obtain representations of a new map from representations of the old one. As an example, we consider the medial. If we are given a representation of M as ρ: V(M)∪ E(M) ∪ F(M) → Rr by row vectors, we can deﬁne         ρ(v) ρ(v) ρ(v) ρ(v) ρ(e) = A ρ(e) and ρ(e) = A ρ(e) . ρ(f ) ρ(f ) ρ(f ) ρ(f )

In practice, we get better results if the new representation ρ is deﬁned from ρ by keeping the vertex part of the representation (that is, ρV = ρV ) and extending ρ to edges and faces so that they are barycentres of their boundary vertices. In 170 Tomaž Pisanski and Arjana Žitnik other situations, we can use spring embedders or other methods to get the vertex part of the representation. There are also other operations, such as the so-called Petrie dual (see [9]), that cannot be described in this way and other tools have to be used to produce meaningful representations for them.

5. Representations of maps in the plane

A planar map can be represented in the plane without crossings, and several algorithms have been developed for this purpose (see, for example, [12], [35]). The algorithms for different convex representations of plane graphs include the Tutte and CaGe methods from Section 3 or the work of Chiba et al. [8]. In this section we suggest how to represent maps on surfaces of arbitrary genus in the plane. Up to now, our definition of the representation of graph was adequate. However, in what follows, that definition becomes too restrictive – we must expand it in order to include some generalizations that seem obvious. For instance, when drawing a graph on a surface in 3-space, the edges are represented as curved lines. Even more interesting is a representation in a polygon, where some edges are represented as unions of line segments. Finally, we introduce representations of graphs whose vertices are circles with two vertices adjacent if and only if the circles touch. To these ends we need to present the theory in a more general form. We begin by extending the definition of representation intuitively, delaying the formal definition to the next section.

Rotational projection The most obvious way to represent an arbitrary map in the plane is the so-called rotational projection of the map: this is the usual drawing of a graph, where the cyclic order of the edges around the vertices (the rotation system) is the same as in the map. It is thus possible to recover the faces of the map from the drawing of its 1-skeleton (see Chapter 1). The graph drawing on the right in Fig. 10 is a rotational projection of K5 on the torus on the left. Any rotational projection can be considered as a graph ‘representation’ in which the edges are represented as curved lines. Amap that lies on a non-orientable surface can be described by a general rotation system in which some of the edges are ﬂat and the others are twisted (see Chapter 1). In the corresponding rotational projection we mark the twisted edges with a cross, indicating that in the process of tracing a face, the local rotation needs to be reversed when we move from one endpoint to the adjacent endpoint along a twisted edge. For example, the map from Fig. 11 has only one face, with boundary 1265326145345612341. 8 Representing graphs and maps 171

5 1 3 4 5 1 3 2 2

Fig. 10. A rotational projection of an embedding of K5 in the torus

1 4

2 3

Fig. 11. A rotational projection of a graph, embedded in a non-orientable surface

In the study of crossing numbers of graphs (see Chapter 7), good drawings of graphs are used, in which no two adjacent edges cross, no two edges cross more than once, no edge crosses itself, no more than two edges cross at a point, and no two edges are tangent. Not every rotational projection can be drawn in the plane as a good drawing – for example, the map with the following rotation scheme for K4 from [1]:

{0: 132, 1: 023, 2: 031, 3: 021}.

(Here a: bcd represents the permutation of edges around the vertex a incident with vertices b, c and d, respectively.) It seems extremely difﬁcult to determine which maps allow rotational projections that are good drawings. So far, the only known algorithm for a general map is systematically to try all the possibilities for crossings of non-adjacent edges. If an edge is crossed by more than one edge, the order of crossings is also important. Crossings can then be considered as new vertices, and by counting faces we can determine whether the graph embedding so obtained is planar; in that case we have a good rotational drawing of the original map. The drawing can then be obtained by some algorithm for drawing plane maps, but the edges being crossed will not be line segments. Since the number of possibilities for crossings grows exponentially in the number of edges, this method works only for small maps. 172 Tomaž Pisanski and Arjana Žitnik

Maps from 2-dimensional graph representations A representation of a graph in R2 with edges represented as line segments can be regarded as a rotational projection of some map. Not all maps admit such a representation, although for planar graphs planar drawings correspond to rotational projections. For general graphs it is of interest to see what kind of embeddings such planar representations deﬁne. One may deﬁne the linear genus as the lowest genus obtained in this way.

Fundamental polygons Another way to represent a map in the plane is to use polygon representations of surfaces, where the surface is represented by a fundamental polygon in which the graph of the map is drawn (see Chapter 1). In such a representation an edge may be represented by a sequence of line segments. Maps can be drawn in fundamental polygons of surfaces in different ways: the degrees of vertices are seen better if all the vertices lie in the interior of the polygon (Fig. 12(a)); the faces are seen better if some vertices lie on the boundary of the polygon and the boundary of the polygon consists entirely of edges of the map (Fig. 12(b)). There is a simple method to obtain the second type of drawing of a given map: choose a spanning tree in the dual map, and then glue two faces of the map along an edge if the corresponding vertices of the dual are adjacent in the spanning tree; then draw this dissected polygon using some algorithm for drawing plane graphs. The ﬁrst type of drawing of a map can be obtained from the drawing of the dual map.

b b 1 1 2 4 4 3 3 a 5 3 a a a 1 5 5 2 2 4 1 1 b b (a) (b)

Fig. 12. Representations of K5 on the torus, different from Fig. 10

Mohar [32], [31] has studied straight-line representations and representations with convex faces in the torus and other ‘ﬂat’ surfaces: the cylinder, the Möbius band and the Klein bottle. He gave characterizations of graphs with convex or straight-line representations on these surfaces, which also led to polynomial-time drawing algorithms. 8 Representing graphs and maps 173

Fundamental polygons of surfaces of higher genus can have many sides, compared with the standard polygons of the same surfaces. Using cutting and gluing operations (such as those deﬁned by Ringel in his pioneering book [41]), they can be reduced to fundamental polygons with fewer sides. But in the resulting drawing some of the edges may cross over the boundary of the polygon many times, making the drawing unclear. Žitnik [51] has studied drawings of graphs in standard polygons of orientable surfaces (those of the a b a−1b−1a b a−1b−1 ... a b a−1b−1 form 1 1 1 1 2 2 2 2 k k k k , which have the smallest possible number of sides), and has characterized maps that can be drawn in a standard polygon so that no edge crosses the boundary of the polygon more than once.

Universal covers Every surface can be ‘unfolded’ to the universal cover. There are three cases: the projective plane and the sphere unfold to the sphere, the torus and Klein bottle unfold to the Euclidean plane, and all other surfaces unfold to the hyperbolic plane. In this process, a map on the surface unfolds as well. The drawings in Fig. 13 were obtained by an algorithm for calculating ‘primal-dual circle packings’ by Mohar [33], where further references on circle packings can be found. In circle packing representations of graphs, the vertices of the graph are represented by circles with disjoint interiors and two vertices are adjacent if and only if the corresponding circles touch, as in Fig. 13(a). We then obtain the drawing in Fig. 13(b) by putting a vertex in the centre of each circle and connecting two adjacent vertices with a line.

(a) (b) Fig. 13. Two representations of Du(BS(G(8, 3))), the dual of the barycentric subdivision of an embedding of the generalized Petersen graph G(8, 3) on the double torus 174 Tomaž Pisanski and Arjana Žitnik

6. Representations of incidence geometries and related topics

Some geometers call a graph in which the vertices are properly coloured an incidence geometry [2]. Formally, an incidence geometry G over a set T is a vertex- coloured graph G = (V, ∼,τ), where V is the set of elements, ∼ is the incidence relation deﬁning the graph, and τ: V → T is the proper vertex-colouring that assigns the type to each element. The incidence graph L(G) = (V, ∼) is known as the Levi graph of the geometry G; this is a generalization of the Levi graph of a map, introduced in Section 4. The number of colours is the rank of the geometry. We let Vt be the set of vertices with colour t.

Simple graphs as rank-2 incidence geometries Each graph has two types of objects, vertices and edges, and the geometry is determined by incidence. The Levi graph of the geometry associated with the graph G is simply its subdivision graph S(G).

Flag-simple maps as rank-3 incidence geometries Each ﬂag-simple map has three types of objects, vertices, edges and faces, and the geometry is determined by their incidences. The Levi graph of the geometry associated with the map M is then the 1-skeleton of the barycentric subdivision of the map. The ﬂags of a map correspond to maximal cliques, in this case the triangles, of the corresponding Levi graph.

Incidence structures Rank-2 incidence geometries are also known as incidence structures. An incidence structure is a triple C = (P,L,I), where P is the set of points, L is the set of lines, and I ⊆ P × L is the incidence relation. The elements of I are called flags. In this case flags correspond to the edges of the associated Levi graph. The correspondence with a rank-2 geometry G is as follows: T ={0, 1}, P = V0, L = V1, I =∼. An incidence structure is lineal if any two distinct lines meet in at most one point; this is equivalent to saying that the Levi graph of a lineal incidence structure has girth at least 6. A (vr ,bk)-combinatorial configuration is a lineal incidence structure C = (P,L,I), such that v =|P |, b =|L|, there are r lines through a point, and there are k points on a line. Note that v = b if and only if r = k, since vr = bk.A (vr ,vr )-configuration is usually shortened to a (vr )-configuration. For example, (73) is the unique combinatorial configuration with the given parameters, known as the Fano plane; its Levi graph is the Heawood graph. The configuration of points and lines arising from the Pappus theorem forms a (93)-configuration (see Fig. 14). 8 Representing graphs and maps 175

Fig. 14. The Pappus graph and the Pappus conﬁguration

Representations of incidence geometries We have seen that representations of graphs are useful for graph drawing purposes and they can be used to help us to visualize graphs. We also considered representations of maps. In both cases we represented vertices ﬁrst and then considered extensions to edges and faces. We can generalize this to incidence geometries. Let G = (V, ∼,τ)and G = (V , ∼,τ) be incidence geometries over a set T . A mapping ρ: V → V , is called a representation of G into G if

• it preserves types: for each v ∈ V , τ (ρ(v)) = τ(v); • it preserves incidence: if v,w ∈ V and v ∼ w, then also ρ(v) ∼ ρ(w).

This general deﬁnition permits a uniform treatment of all the previously mentioned representations.

Example 4 Let G be a graph and G = (V, ∼,τ)be the corresponding incidence V = 2 geometry, where the vertices have type 1 and the edges have type 2. Let 1 R V ={λa + µb: a,b ∈ V ,λ,µ≥ ,λ+ µ = } and 2 1 0 1 be the set of line segments 2 G = (V , ∼,τ) V = V ∪ V in R . Now deﬁne an incidence geometry , where 1 2 a ∈ V ,p ∈ V a ∼ p a p τ: V →{ , } and, for 1 2, let if is an endpoint of . Deﬁne 1 2 , a ∈ V → p ∈ V → ρ: V → V 1 1, 2 2. Then any representation of the incidence geometry G into the incidence geometry G is just a planar graph representation.

Example 5 Let M be a ﬂag-simple map and let G = (V, ∼,τ)be the corresponding incidence geometry where the vertices have type 1, the edges have type 2 and the faces have type 3. We deﬁne the incidence geometry G = (V , ∼,τ) into which we will represent the map M.Asimple curve in R3 is the image of a one-to-one continuous function f :[0, 1]→R3; the points f(0) and f(1) are the endpoints of the curve. A disc in R3 is the image of a one-to-one continuous 176 Tomaž Pisanski and Arjana Žitnik

g:[ , ]×[ , ]→ 3 S 3 V S function 0 1 0 1 R . Let be a surface in R . Define 1 to be , V S V S 2 to be the set of simple curves on , and 3 as the set of discs on ; then V = V ∪ V ∪ V a ∈ V p ∈ V a 1 2 3. Any element 1 is incident to 2 if is an endpoint p p ∈ V ∪ V f ∈ V p of , and 1 2 is incident to 3 if belongs to the boundary of f τ : V →{, , } a ∈ V → p ∈ V → f ∈ V → . Define 1 2 3 by 1 1, 2 2 and 3 3. Suppose that a representation ρ: V → V of the incidence geometry G into G has the following property: the vertices map to distinct points in S and the images of edges are disjoint except for common vertices of adjacent edges. Then ρ is an embedding of the map M in the surface S. Example 6 The real projective plane P is an incidence structure defined as follows. The points of P are 1-dimensional subspaces of R3 and the lines of P are 2-dimensional subspaces of R3. Instead of labelling elements of the projective plane with subspaces of R3, we can use homogeneous coordinates and label them with vectors of R3\{(0, 0, 0)}. Then a point p (that is, the line through the origin with direction p) lies on a line (that is, the plane through origin with normal vector ) if and only if pT = 0. The projective plane is also a rank-2 incidence geometry. We can represent incidence structures in the projective plane by using homogeneous coordinates: we call such a representation a homogeneous representation. It is perhaps not surprising that the homogeneous representations of incidence structures are related to the problem of drawing the structures with straight lines. In particular, we are interested in (v3) configurations. A homogeneous representation is ρ strong if ρ(p)Tρ() = 0 implies that p lies on , and a homogeneous representation that is not strong is weak. In his 2 dissertation [44], Steinitz proved that each (v3)-configuration can be drawn in R with at most one curved line. His result can be rephrased in modern terminology as follows:

Theorem 6.1 An incidence structure that is obtained from a (v3)-conﬁguration by removal of a single line admits a weak homogeneous representation.

Take for instance the Fano plane, the unique smallest (73)-conﬁguration, whose Levi graph is the Heawood graph H. It is well known that no real 3-dimensional homogeneous representation exists for H (or, equivalently, for the Fano plane). However, if we delete any vertex of H then the resulting graph admits such a representation, and this in turn can be interpreted as a drawing of the incidence structure so obtained with straight lines in the Euclidean plane. On the other hand, the Pappus graph does admit a real homogeneous representation (see Fig. 15). The solid vertices of the Pappus graph can be interpreted as points and the vectors them are in parentheses; the hollow vertices can be interpreted as lines, and their representing vectors are put in square brackets. By a suitable projection to the plane, we obtain a straight-line drawing of the Pappus conﬁguration in the plane. 8 Representing graphs and maps 177

[0,1,–2]

(–2,2,1) (0,2,1) (2,2,1) [2,1,2] (0,–2,1) [–2,1,2] [2,1,–2] [2,–1,2] (–1,0,1) (1,0,1) (2,–2,1) [0,1,2] (–2,–2,1) [1,1,0] [0,1,0] [–1,1,0]

(0,0,1)

Fig. 15. The Pappus graph

Related topics There are many topics related to representation of graphs that we have no space to discuss. We mention in passing that frameworks from rigidity theory are graph representations (see Graver, Servatius and Servatius [21]) and that representations give rise to connectivity algorithms (see Linial, Lovasz and Widgerson [29]). Visibility is another example (see, for example, Tamassia and Tollis [46]). Representations of graphs also arise as an important tool in the study of distance- regular graphs (see Brouwer, Cohen and Neumaier [7]). Here, brieﬂy noted, are two other topics.

The representation polytope Given a representation ρ of a graph G, the convex hull of the set {ρ(v): v ∈ V (G)} is a polytope P(ρ), called the representation polytope of G. We denote its graph by ρ(G). A representation polytope is an eigenpolytope of G if the vectors representing the vertices form an orthonormal basis of the eigenspace corresponding to some eigenvalue of the adjacency matrix of G. Godsil [18] proved the following remarkable result. Theorem 6.2 Let G be a distance-regular graph, and let P be the eigenpolytope associated with the second-largest eigenvalue of G. Then G is the graph of P if and only if it is one of the following: a Johnson graph J(v,k), a Hamming graph H (n, k), a halved n-cube, the Schläﬂi graph, the Gosset graph, the icosahedron, the dodecahedron, a generalized octahedron Kr(2) or a cycle Cn.

Using graph structure to improve representations Sometimes the graphs we want to draw possess some structure; for example, they might be symmetric, be products of smaller graphs, or belong to some special class. Knowing something about the structure of a graph, we can usually obtain a better drawing of the graph that also reveals some of that structure. We illustrate this approach with an example. 178 Tomaž Pisanski and Arjana Žitnik

Consider the graph G obtained from the hypercube Q4 in Fig. 16(c) by removing the two dashed edges; the drawing in Fig. 16(a) was obtained by a standard spring- embedder algorithm in 3-space. Fig. 16(b) shows a much better drawing, one obtained by ﬁrst recognizing that G is a subgraph of Q4, determining the vertex coordinates of Q4 in 4-space, and then choosing an appropriate projection to the plane. The computer system vega [37] uses this idea in order to ﬁnd good drawings of various graph products, line graphs, tilings, etc. For a standard introduction to graph products the reader may consult the book by Imrich and Klavžar [25], while Grünbaum and Shephard’s monumental monograph [22] provides a detailed treatment of tilings.

(a) (b) (c)

Fig. 16. Two drawings of G and a drawing of the hypercube Q4

Several algorithms exploit symmetry of graphs to obtain pleasing drawings of graphs – see, for example, [24], where Hong, McKay and Eades gave a linear- time algorithm for constructing maximally symmetric straight-line drawings of 3-connected planar graphs.

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JIN HO KWAK and JAEUN LEE

1. Introduction 2. Graph coverings 3. Regular coverings 4. Surface branched coverings 5. Regular surface branched coverings 6. Distribution of surface branched coverings 7. Further remarks References

Many graphs with a symmetry property can be described as coverings of simpler graphs. In this chapter, we survey several enumeration problems for various types of non-isomorphic graph coverings of a graph and of surface branched coverings. We also discuss the distribution of surface branched (regular) coverings of a surface through the genera of supporting surfaces.

1. Introduction K Consider the map from the 3-dimensional cube Q3 to the complete graph 4, deﬁned by identifying antipodal points of the cube. The vertices and edges of the cube map two-to-one to the vertices and edges of the complete graph. However, at each vertex and its incident edges the mapping is one-to-one. This quotient map is an example of a covering projection. Loosely speaking, a covering projection is a surjective graph homomorphism that is an isomorphism when restricted to each vertex and its incident edges. For a formal deﬁnition of the covering projection, let G be a connected simple graph. The neighbourhood N(v) of a vertex v ∈ V (G) is the set of vertices adjacent to v. A map p: G → G is a covering projection if p|N(v)˜ : N(v)˜ → N(v)

181 182 Jin Ho Kwak and Jaeun Lee is a bijection for any vertex v ∈ V (G) and v˜ ∈ p−1(v). The image graph G is the base graph, and the domain graph G is a covering graph. We call p: G → G a covering of G. When the covering projection p is r-to-one, p: G → G is an r-fold covering. In particular, a 1-fold covering projection is a graph isomorphism. A covering p: G → G is regular,oraΓ -covering, if there is a subgroup of the automorphism group Aut (G) of G acting freely on G so that h ◦ p = g, where g: G → G/ is the quotient map and h: G → G/ is a graph isomorphism. Two coverings p1: G1 → G and p2: G2 → G are isomorphic if there exists a graph isomorphism α: G1 → G2 such that p2 ◦ α = p1; we call α a covering isomorphism. In particular, when G1 = G2 (= G, say) with p1 = p2 (= p, say), α is a covering transformation, and the set of all covering transformations forms a group under composition, called the covering transformation group of the covering p: G → G. Next, we introduce a construction method for graph coverings (see Gross and Tucker [7]). Each edge of a graph G gives rise to a pair of opposite arcs, and we designate one of them as positive and the other as negative. We denote the resulting set of arcs of G by A(G).Apermutation voltage assignment φ of a graph G is − − a map φ: A(G) → Sr with the property that φ(e 1) = φ(e) 1 for each arc e in A(G), where Sr is the symmetric group on the set {1, 2,...,r}. The permutation derived graph Gφ is defined as follows: V(Gφ) = V (G) ×{1, 2,...,r} and E(Gφ) = E(G) ×{1, 2,...,r}, where for each edge e joining v and w and each j ∈{1, 2,...,r},(e,j)is an edge joining (v, j) and (w, φ(vw)j). Then the natural projection p: Gφ → G is an r-fold covering. Let be a finite group. An ordinary voltage assignment (or -voltage assignment)ofG is a function φ: A(G) → with the property that φ(e−1) = − φ(e) 1 for each arc e. Let S denote the symmetric group on the elements of . The (left) regular representation of is the map L: → S defined by g → L(g), the left multiplication by g on ; this representation is faithful and the group can be identified with the group of left translations L(g): ≡{L(g): g ∈ }; this is Cayley’s theorem. Note that a permutation voltage assignment φ: A(G) → S with its images in is just a -voltage assignment of G, and the ordinary derived graph G ×φ derived from a -voltage assignment φ: A(G) → is the φ permutation derived graph G . Notice that the natural projection p: G ×φ → G is a -covering. Gross and Tucker ([7], [8, Chap. 2]) showed that every covering of a graph G can be derived from a permutation voltage assignment φ that assigns the identity to the arcs of some spanning tree of G. Furthermore, if the covering is regular, then the voltage assignment is ordinary. A covering of a graph with loops or multiple edges can be constructed similarly by adding extra vertices on each loop and on each multiple edge as necessary. 9 Enumerating coverings 183

Fig. 1. Two non-isomorphic coverings with the same covering graph

We note that a graph G can cover the base graph G in several different ways. Fig. 1 gives two non-isomorphic regular coverings of the bouquet of two circles. The covering graph on the left uses the dihedral group D4 as its voltage group, while that on the right uses Z2 × Z4. In this chapter, we are interested in enumerating covering projections, rather than the covering graphs, and the numbers computed always refer to non- isomorphic coverings.

2. Graph coverings

Let T be a spanning tree of a connected graph G. A permutation voltage assignment φ is T -normalized if φ assigns the identity to the arcs of T . Let CT (G; r) be the set of all T -normalized permutation voltage assignments φ: A(G) → Sr of G. In order to find an algebraic property when two r-fold coverings p: Gφ → G and q: Gψ → G are isomorphic, we assume that they are isomorphic by a covering isomorphism α: Gφ → Gψ . Then, for each v ∈ V (G), −1 −1 α|p−1(v): p (v) → q (v) is a permutation on the r vertices {v1,v2,...,vr }, and can be considered as a permutation on the set {1, 2,...,r}. We define f : V (G) → Sr by f(v)= α|p−1(v) for v ∈ V (G), so that α(v,h) = (v, f (v)(h)) for each vertex (v, h) of the covering graph Gφ. For each arc vw of G,if(v, h) is adjacent to (w, k) in Gφ, then φ(vw)(h) = k, and α(v,h) = (v, f (v)(h)) is adjacent to α(w,k) = (w, f (w)(k)) in Gψ . Thus, ψ(vw)f (v) = f(w)φ(vw), 184 Jin Ho Kwak and Jaeun Lee or ψ(vw) = f (w)φ(uv)f (v)−1, for all vw ∈ A(G). In [22] we showed that the converse is also true. Theorem 2.1 Two r-fold coverings p: Gφ → G and q: Gψ → G are isomorphic if and only if there exists a function f : V (G) → Sr such that ψ(vw) = − f (w)φ(vw)f (v) 1, for each arc vw.Ifφ,ψ ∈ CT (G; r), then this is equivalent − to saying that there exists a permutation σ ∈ Sr such that ψ(vw) = σ φ(vw)σ 1, for each arc vw not in T . Let Iso (G; r) denote the number of r-fold coverings of a graph G, and let β = β(G) =|E(G)|−|V (G)|+1 be the cycle rank of G. Then we can consider a normalized permutation voltage assignment as a β-tuple of permutations in Sr by labelling the positive arcs in A(G) − A(T ) as e1,e2,...,eβ , and the set CT (G; r) can be identified as

CT (G; r) = Sr × Sr ×···×Sr (with β factors).

We deﬁne an Sr -action on the set CT (G; r) by simultaneous coordinate-wise conjugacy: for any σ ∈ Sr and any (σ1,σ2,...,σβ ) ∈ CT (G; r),

−1 −1 −1 σ(σ1,σ2,...,σβ ) = (σ σ1σ ,σσ2σ , ... ,σσβ σ ).

It follows from Theorem 2.1 that two normalized permutation voltage assignments φ and ψ in CT (G; r) yield isomorphic coverings of G if and only if they belong to the same orbit under the Sr -action. We deduce the following theorem from the orbit-counting theorem (Burnside’s Lemma). Theorem 2.2 The number Iso (G; r) of r-fold coverings of G is β−1 2 r Iso (G; r) = 1! 2!2 ···r !r . 1+22+···+rr =r

Table 1 gives the numbers Iso (G; r) for small r and β.

Table 1. The numbers Iso (G; r) for small r and β

βr= 1 r = 2 r = 3 r = 4 r = 5

11 2 3 5 7 2 1 4 11 43 161 3 1 8 49 681 14721 4 1 16 251 14491 1730861 5 1 32 1393 336465 207388305 6 1 64 8051 7997683 24883501301 9 Enumerating coverings 185

Next, we compute the number Isoc (G; r)of connected r-fold coverings of G.It is well known (see, for example, Massey [30]) that the fundamental group of a graph G is a free group of rank β, and that there is a one-to-one correspondence between the connected r-fold coverings of G and the conjugacy classes of subgroups of index r in the fundamental group of G. Let F(β) denote the free group generated by β elements and let αF(β)(r) be the number of subgroups of F(β) of index r. This number was determined by Hall [9]:

r−1 β−1 β−1 αF(β)(r) = r(r!) − ((r − j)!) αF(β)(j), with αF(β)(1) = 1. j=1

Liskovets [27] computed the number Isoc (G; r)of conjugacy classes of subgroups of F(β) of index r in terms of the numbers αF(β)(m) for the divisors m of r. Theorem 2.3 The number Isoc (G; r) is given by r (G; r) = 1 α (m) µ d(β−1)m+1, Isoc r F(β) md m|r d|(r/m) where µ is the number-theoretic Möbius function. Table 2 gives the values of Isoc (G; r) for small r and β.

Table 2. The numbers Isoc (G; r) for small r and β

βr= 1 r = 2 r = 3 r = 4 r = 5 r = 6

11 1 1 1 1 1 2 1 3 7 26 97 624 3 1 7 41 604 13753 504243 4 1 15 235 14120 1712845 371515454 5 1 31 1361 334576 207009649 268530771271 6 1 63 7987 7987616 24875000437 193466859054994

Later, another formula for the number of connected r-fold coverings of a graph was found independently by Hofmeister [13] and by the authors [23].

3. Regular coverings

Let CT (G; ) be the set of T -normalized -voltage assignments of G.Any regular r-fold covering of G is isomorphic to an ordinary derived covering p: G×φ → G for a group of order r and a normalized ordinary voltage assignment φ in 186 Jin Ho Kwak and Jaeun Lee

CT (G; ) (see Gross and Tucker [7]). Moreover, if the graph G×φ is connected, then the group becomes the covering transformation group. The algebraic characterization of two isomorphic graph coverings given in Theorem 2.1 can be rephrased for regular coverings as follows (see Hong et al. [16] and Skoviera˘ [37]).

Theorem 3.1 Let φ,ψ ∈ CT (G; ) be -voltage assignments of G that yield connected coverings. Then their derived coverings are isomorphic if and only if there exists a group automorphism σ ∈ Aut() such that ψ(vw) = σ(φ(vw))for all arcs vw in G but not in T .

It is not difficult to show that the components of any regular covering G×φ → G are isomorphic as coverings of G, and that any two isomorphic connected regular coverings of G must have isomorphic covering transformation groups. Moreover, any two regular coverings of G of the same fold number are isomorphic if and only if their components are isomorphic as coverings. Note that each component of a -covering of G is an S-covering of G for some subgroup S of . We now let Iso (G; ) be the number of regular -coverings and let Isoc (G; ) be the number that are connected. Similarly, we let IsoR(G; r) be the number of regular r-fold coverings, regardless of the group involved, and let IsocR(G; r) be the number that are connected. The following theorem of Kwak, Chun and Lee [20] lists some basic counting properties about regular coverings. ( ) r ∈ R(G; r) = R(G; d) Theorem 3.2 a For any N, Iso d|r Isoc . R (b) For any r ∈ N, Isoc (G; r) = Isoc (G; ), where the sum is over all r non-isomorphic groups of order . (c) For any finite group , Iso (G; ) = S Isoc (G; S), where the sum is over all non-isomorphic subgroups of . (d) For any two finite groups 1 and 2 with (|1|, |2|) = 1, Iso (G; 1 ⊕ 2) = Iso (G; 1) Iso (G; 2), and Isoc (G; 1 ⊕ 2) = Isoc (G; 1) Isoc (G; 2), where 1 ⊕ 2 is the direct sum of 1 and 2. To complete our enumeration of regular coverings, we need to find computational formulas for Isoc (G; ). Note that the set CT (G; ) of normalized -voltage assignments of G can be identified as

CT (G; ) = × ×···×(with β factors).

Let

r (; r) ={(g1,g2,...,gr ) ∈ :{g1,g2,...,gr } generates }. 9 Enumerating coverings 187

Then a β(G)-tuple g = (g1,g2,...,gβ ) is associated with a connected -covering if and only if g ∈ (; β). By Theorem 3.1, under the coordinate-wise Aut ()- action on the set (; β), any two elements in (; β)belong to the same orbit if and only if they yield isomorphic connected -coverings. Since the Aut ()-action on (; β) is free, we can deduce the following theorem from the orbit-counting theorem.

Theorem 3.3 For any ﬁnite group ,

| (; β)| Isoc (G; ) = . |Aut ()|

To complete the enumeration of IsoR(G; r) and IsocR(G; r), we need to determine |Aut ()| and | (; β)| for any finite group . When is Abelian, this was done by Kwak, Chun and Lee [20]. Next, we introduce a formula to compute the number | (; β)| for any finite group in terms of the Möbius function defined on the subgroup lattice of . The Möbius function assigns an integer µ(K) to each subgroup K of by the recursive formula 1ifK = , µ(H) = K<. H≥K 0if Jones (see [18], [19]) used such a function to count the normal subgroups of a surface group or a crystallographic group, and applied it to count certain covering surfaces. We see that β || =|CT (G; )|= | (K; β)|. K≤

It follows from Möbius inversion that β | (; β)|= µ(K)|CT (G; K)|= µ(K)|K| . K≤ K≤

The next theorem can be deduced from Theorem 3.3.

Theorem 3.4 For any ﬁnite group , (G; ) = 1 µ(K)|K|β . Isoc | ()| Aut K≤

The cyclic group = Zr has a unique subgroup Zd for each d dividing r, and has no other subgroups. The Möbius function on the subgroup is µ(Zd ) = 188 Jin Ho Kwak and Jaeun Lee

µ(r/d) (the number-theoretic Möbius function) and |Aut (Zr )|=ϕ(r) (the Euler ϕ-function), so this implies that 1 r β (G; Zr ) = µ d . Isoc ϕ(r) d d|r

The following result gives Iso (G; Zr ) and Isoc (G; Zr ) in terms of the prime factorization of r. s s s r = p 1 p 2 ···p ≥ Theorem 3.5 If 1 2 2, then the total number of Zr -coverings of G is   β =  1if0,    (si + 1) if β = 1, Iso (G; Zr ) =  i=1   (pβ − )(psi (β−1) − )  i 1 i 1  1 + if β ≥ 2 , (p − )(pβ−1 − ) i=1 i 1 i 1 and the number of connected Zr -coverings of G is   0ifβ = 0,  β Isoc (G; Zr ) = p −  (β−1)(si −1) i 1  pi if β ≥ 1. pi − 1 i=1

For the dihedral group Dr of order 2r, a similar computation gives the following [20]:

Theorem 3.6 For r ≥ 3, the number of connected Dr -coverings of G is

 pβ−1 − β (mi −1)(β−2) i 1 Isoc (G; Dr ) = 2 − 1 pi , pi − 1 i=1

m m r = p 1 ···p where 1 . Even though we have a general computational formula for Isoc (G; ),itis not easy to ﬁnd its exact value if the lattice of subgroups of is complicated. However, if is Abelian, we can derive an enumeration formula for Isoc (G; ) that does not involve the lattice structure of the subgroups of . Our next result follows from Theorem 3.3. 9 Enumerating coverings 189

Theorem 3.7 Let t1,t2,...,t and s1,s2,...,s be natural numbers with s1 > s > ···>s ⊕ t s G 2 . Then the number of connected h=1 hZp h -coverings of is

t β−i+ p 1 − 1

 f(β,ti ,si ) i=1 Isoc (G;⊕h= thZpsh ) = p , 1 tj t −h+ p j 1 − 1 j=1 h=1 where p is prime, t = t1 + t2 +···+t and

 −1 f(β,ti,si) = (β − t) ti(si − 1) + ti tj (si − sj − 1). i=1 i=1 j=i+1

The number Iso (G; ) can be computed for any ﬁnite Abelian group by using Theorems 3.2 and 3.7, repeatedly if necessary. Table 3 shows the numbers Iso (G; ) and Isoc (G; ) for some Abelian groups and small β.

Table 3. The numbers Iso (G; ) and Isoc (G; ) for some and small β

Iso Isoc

β(p,q)Zp3 ⊕ Zp Zq2 Zp3 ⊕ Zp ⊕ Zq2 Zp3 ⊕ Zp Zq2 Zp3 ⊕ Zp ⊕ Zq2

1 (2,3) 4 3 12 0 1 0 2 (2,5) 32 37 1184 6 30 180 3 (3,5) 2757 807 2224899 1404 775 1088100 4 (3,7) 161451 137601 22215819051 126360 137200 1695792000

Corollary 3.8 For any t ≥ 1, the total number of tZp-coverings of G is

t (pβ − 1)(pβ−1 − 1) ···(pβ−h+1 − 1) Iso (G; tZp)=1 + , (ph − 1)(ph−1 − 1) ···(p − 1) h=1 and the number of connected tZp-coverings of G is

(pβ − 1)(pβ−1 − 1) ···(pβ−t+1 − 1) Isoc (G; tZp) = . (pt − 1)(pt−1 − 1) ···(p − 1)

Again, we note that Hofmeister [11] independently computed the number Isoc (G; tZp) by another method. 190 Jin Ho Kwak and Jaeun Lee

We note two facts:

• the number of connected mZp-coverings of a connected graph G is equal to the number of the m-dimensional subspaces of the β-dimensional vector space over the ﬁeld GF (p) (see [38, Section 1.4]); • for a connected -covering p: G → G, the image p∗(π1(G)) of the fundamental group of the covering graph G is a normal subgroup of the fundamental group π1(G) of the base graph G, and the quotient group π1(G)/p∗(π1(G)) is isomorphic to .

Now, it is not hard to show that the number

| (; β)| (G; ) = , Isoc | ()| Aut where runs over all non-isomorphic Abelian groups of order r, is equal to the number of subgroups of index r in the free Abelian group Z ⊕···⊕Z generated by β elements. Table 4 shows these numbers for small r and β.

r ⊕β Table 4. The number of subgroups of index in 1 Z

βr= 12345678

111111111 21347612815 3171335319157155 4 1 15 40 155 156 600 400 1395 5 1 31 121 651 781 3751 2801 11811 6 1 63 364 2667 3906 22932 19608 97155 7 1 127 1093 10795 19531 138811 137257 788035 8 1 255 3280 43435 97656 836400 960800 6347715 9 1 511 9841 174251 488281 5028751 6725601 50955971 10 1 1023 29524 698027 2441406 30203052 47079208 408345795

Given the classiﬁcation of the groups of order r, IsoR(G; r) and IsocR(G; r) can be computed by Theorem 3.2. For example,

R 2 Iso (G; p ) = Isoc (G; 2Zp) + Isoc (G; Zp2 ) + Isoc (G; Zp) + 1 β β−1 β β (p − 1)(p − 1) (β− ) p − 1 p − 1 = + p 1 + + 1. (p2 − 1)(p − 1) p − 1 p − 1

For small values of r and β, these numbers are given in Tables 5 and6. 9 Enumerating coverings 191

Table 5. The numbers IsoR(G; r) for small r and β

βr= 12 3 4 5 6 7 8 9 10 11

112232424342 2 1 4 5 11 7 23 9 30 18 31 13 3 1 8 14 43 32 140 58 254 144 298 134 4 1 16 41 171 157 851 401 2126 1251 2977 1465 5 1 32 122 683 782 5144 2802 17452 11133 29860 16106

Table 6. The numbers IsocR(G; r) for small r and β βr= 123456 7 8 9 1011

111111111111 213476158 19132112 3 1 7 13 35 31 119 57 211 130 259 133 4 1 15 40 155 156 795 400 1955 1210 2805 1464 5 1 31 121 651 781 4991 2801 16771 11011 29047 16105

4. Surface branched coverings

Let S be a closed surface. A continuous surjective map p: S˜ → S is a branched ˜ −1 covering if p|S˜−p−1(B): S − p (B) → S − B is a covering for a ﬁnite subset B of S. The smallest subset B of S with this property is the branch set. A branched ˜ ˜ −1 covering p: S → S is regular,ora-covering,ifp|S˜−p−1(B): S − p (B) → S − B is a regular covering, with the covering transformation group .Two ˜ ˜ branched coverings p1: S1 → S and p2: S2 → S are isomorphic if there exists a ˜ ˜ ˜ ˜ homeomorphism h: S1 → S2 for which p2 ◦ h = p1. For a combinatorial description of branched surface coverings, Hurwitz [14] introduced a system (now called a Hurwitz system), and Gross and Tucker [7] introduced embedded voltage graphs. In fact, the Hurwitz system is a kind of voltage assignment on a bouquet of circles (see Kwak et al. [21]).˙ Let be the 1 bouquet of circles and let |B|=b. Let C ( 2h+b → Sh − B; r) be the subset of 2h+b (Sr ) consisting of all (2h+b)-tuples (σ1,σ2,...,σ2h+b), (that is, permutation voltage assignments) that satisfy the following three conditions:

(A1) The subgroup of Sr generated by {σ1,σ2,...,σ2h+b} is transitive on {1, 2,...,r}. h b −1 −1 (A2) σiσh+iσi σh+i σ2h+i = 1. i=1 i=1 (A3) σi = 1 for each i = 2h + 1, 2h + 2,...,2h + b. 192 Jin Ho Kwak and Jaeun Lee

The Hurwitz existence and classiﬁcation theorem can be rephrased as follows.

Theorem 4.1 Let S be an orientable surface Sh, and let B be any set of b points 1 in S. Then each permutation voltage assignment φ in C ( 2h+b → S − B; r) φ induces a connected branched r-fold covering p˜φ: S → S of S with branch set B. Conversely, every connected branched r-fold covering of S with branch set B 1 can be derived from a voltage assignment in C ( 2h+b → S − B; r). Furthermore, for two such permutation voltage assignments φ1 and φ2, φ φ p˜ : S 1 → S p˜ : S 2 → S the derived branched coverings φ1 and φ2 are p p isomorphic if and only if the corresponding graph coverings φ1 and φ2 are isomorphic.

A corresponding result holds for the non-orientable surface Nk: the only difference is that k+b replaces 2h+b and (A2) is replaced by k b

(A2) σiσi σk+i = 1. i=1 i=1 In fact, condition (A1) guarantees that the surface Sφ is connected, and the conditions (A2) (or (A2) ) and (A3) imply that the set B is the same as the branch φ set of the branched covering p˜φ: S → S. Mednykh [32], [36] enumerated the conjugacy classes of subgroups of index r in the fundamental group of any surface S. To state his results, we need some further notation. Let αs(m) denote the number of subgroups of index m in the fundamental group π1(S, ∗), where s is the number of handles or crosscaps of the S α+(m) = α ( 1 m) m surface , depending on the orientability. Also, let s s 2 if is even − + and0ifm is odd, and let αs (m) = αs(m) − αs (m).

Theorem 4.2 The number of connected r-fold unbranched coverings of a surface Sh or Nk is r (S ,φ; r) = 1 α (m) µ d(2h−2)m+2; Isoc h r h md m|r d|(r/m)

n (N ,φ; r) = 1 µ d(k−2)m+1( ( ,d)α−(m) + dα+(m)). Isoc k r md gcd 2 k k m|r d|(r/m)

An explicit formula for αk(m) can be found in [32] and [36]. By using Theorem 4.1, we can express the number Isoc (S, B; r) of connected r-fold branched coverings of the surface S with branch set B in terms of known parameters (see [26]). 9 Enumerating coverings 193

Theorem 4.3 Let S be either Sh or Nk and let B be a set of b points in S. Then the number of connected r-fold branched coverings of S with branch set B is

b− 1 b (S ,B; r) = (− )b (S ,φ; r) + (− )t ( ; r), Isoc h 1 Isoc h 1 t Isoc 2h+b−t−1 t=0

b− 1 b (N ,B; r) = (− )b (N ,φ; r) + (− )t ( ; r), Isoc k 1 Isoc k 1 t Isoc k+b−t−1 t=0 where m is the bouquet of m circles.

5. Regular surface branched coverings

In this section, we return to regular coverings. For a combinatorial description of regular surface branched coverings of an orientable surface, let be a ﬁnite 1 2h+b group. Let C ( 2h+b → Sh − B; ) be the subset of () consisting of all (2h + b)-tuples (g1,g2,...,g2h+b) (that is, ordinary voltage assignments) that satisfy the following three conditions:

(B1) The subgroup generated by {g1,g2,...,g2h+b} is the full group . h b −1 −1 (B2) gigh+igi gh+i g2h+i = 1. i=1 i=1 (B3) gi = 1 for each i = 2h + 1, 2h + 2,...,2h + b.

Acorresponding result holds for the non-orientable surface Nk: the only difference is that k+b replaces 2h+b and (B2) is replaced by k b

(B2) gigi gk+i = 1. i=1 i=1 Kwak, Kim and Lee [21] obtained a Hurwitz-like existence and classiﬁcation theorem on regular branched surface coverings.

Theorem 5.1 Let S be an orientable surface Sh, let B be any set of b points in S, and let be a ﬁnite group. Then each ordinary voltage assignment φ in 1 φ C ( 2h+b → S − B; ) induces a connected branched -covering p˜φ: S → S of S with branch set B. Conversely, every connected branched -covering of S with branch set B can 1 be derived from a voltage assignment in C ( 2h+b → S − B; ). Furthermore, for two such ordinary voltage assignments φ1 and φ2, the derived φ φ p˜ : S 1 → S p˜ : S 2 → S branched coverings φ1 and φ2 are isomorphic if and only p p if the corresponding graph coverings φ1 and φ2 are isomorphic. 194 Jin Ho Kwak and Jaeun Lee

A corresponding result holds for the non-orientable surface Nk: the only 1 1 difference is that C ( k+b → S − B; ) replaces C ( 2h+b → S − B; ). For a ﬁnite group , let Isoc (S, B; ) be the number of connected branched -coverings of the surface S with branch set B. Note that if two connected regular branched coverings are isomorphic, then their covering transformation groups (or voltage groups) should be isomorphic. The following equation comes from the fact that every connected regular r-fold branched covering is a connected branched -covering for some group of order r: R Isoc (S, B; r) = Isoc (S, B; ), where the sum is over all isomorphism classes of groups of order r. By reasoning similar to that for Theorem 4.3, we can use unbranched coverings to enumerate branched ones (see Kwak et al. [26]).

Theorem 5.2 Let S be either Sh or Nk, let B be a set of b points in S, and let be a ﬁnite group. Then the number of connected branched -coverings of S with branch set B is b− 1 b (S ,B; ) = (− )b (S ,φ; ) + (− )t ( ; ), Isoc h 1 Isoc h 1 t Isoc 2h+b−t−1 t=0

b− 1 b (N ,B; ) = (− )b (N ,φ; ) + (− )t ( ; ), Isoc k 1 Isoc k 1 t Isoc k+b−t−1 t=0 where m is the bouquet of m circles. Kwak, Lee and Mednykh [26] enumerated branched -coverings of a surface. To state their result, we need yet more notation. For an irreducible character η of , let   1ifρ is real, c = 1 η(g2) = − η ρ η ||  1ifis real but is not real, g∈ 0ifη is not real, with the representation ρ corresponding to η. By combining Theorem 3.4 and the result in [18] and [32], we can rephrase Theorem 5.2 as follows.

Theorem 5.3 Let S be either Sh or Nk, let B be a set of b points in S and let be a ﬁnite group. Then the number of connected branched -coverings of S with branch set B is µ(K) |K|2h−1 (S ,B; ) = (|K|− )b + (− )b η( )2−2h , Isoc h | ()| 1 1 1 K≤ Aut η 9 Enumerating coverings 195

µ(K) |K|k−1 (N ,B; ) = (|K|− )b + (− )b ckη( )2−k , Isoc k | ()| 1 1 η 1 K≤ Aut η where η ranges over all irreducible characters of K except the principal character. We observe that if is a finite Abelian group, then every irreducible character is of degree 1. So, for each subgroup K of , 2−2h k 2−k λ(K) η(1) =|K|−1 and cηη(1) = 2 − 1, η η where the sum is over all irreducible characters of K and λ(K) is the number of direct summands of K whose orders are even. Note also that Theorem 5.3 is efficient if the lattice structure of subgroups of and their characters are known, as in the case of regular graph coverings. Enumerating formulas for Isoc (Sh,φ; ) and Isoc (Nk,φ; ) for any finite Abelian group that do not involve the lattice structure of subgroups of are given in [26]. By using this and Theorem 3.7, we can obtain another enumeration formula from Theorem 5.2 that does not involve the lattice structure of subgroups of .

6. Distributions of surface branched coverings

For any two surfaces S and S˜, let Iso (S, B; S˜; r)be the number of r-fold branched coverings p: S˜ → S with branch set B, and let IsoR(S, B; S˜; r) be the number of those that are regular. Similarly, let Iso (S, B; S˜; ) be the number of branched -coverings p: S˜ → S with branch set B. Mednykh [34] obtained a formula for counting the connected branched coverings of an orientable surface with arbitrarily prescribed ramiﬁcation type. From this, we can compute the number Iso (Sh,B; S; r)for any orientable surface Sh and any r, because every branched covering of an orientable surface is also orientable. However, the numbers Iso (Nk,B; S; r) are not yet completely determined. In contrast, the numbers IsoR(S, B; S˜; r) are completely determined. It is clear that Iso (S, B; S˜; 2) = IsoR(S, B; S˜; 2). For a prime number p, Kwak, S. Kim, Lee and J. Kim [21], [24], [29] computed the numbers IsocR(S, B; S˜; p), ˜ ˜ Isoc (S, B; S; Dp) and Isoc (S, B; S; mZp). For more general cases, Jones [18], [19] obtained a counting formula for the connected branched -coverings of a surface with a prescribed ramiﬁcation type; using this formula, we can derive computational formulas for the numbers Iso (Si,B; Sj ; ) and Iso (Ni,B; Nj ; ) when || is odd. Kwak, Lee and Mednykh [25] obtained a formula that counts the connected branched orientable -coverings of a non-orientable surface when 196 Jin Ho Kwak and Jaeun Lee

is Abelian. Goulden, Kwak and Lee [5] obtained a formula for the number of connected branched orientable -coverings of a non-orientable surface with a prescribed ramiﬁcation type; this gives computational formulas for the numbers R Iso (Ni,B; Sj ; )and Iso (Ni,B; Sj ; r). Note that we can obtain a computational formula for IsoR(S, B; S˜; r) for any two surfaces S and S˜ by combining this and the results in Jones [18], [19]. Here, we present the results only in the prime case.

Theorem 6.1 Let B be a b-subset of a surface S and let p be a prime. If S = Sh, then   (p2h − )/(p − ) S˜ = S b = ,  1 1 if 1+p(h−1) and 0   p2h−1((p − )b−1 + (− )b) R(S ,B; S˜; p) = 1 1 Iso h  S˜ = S b = ,  if ph+ 1 (p−1)(b−2) and 0  2  0 otherwise.

If S = Nk, then   S˜ = S b = ,  1ifk−1 and 0   k − S˜ = N k = ,b= , R ˜ 2 2if2(k−1) and 1 0 Iso (Nk,B; S; 2) =  k S˜ = N b = ,b ,  2 if 2(k−1)+b and 0 even  0 otherwise, and for an odd prime p,  k−1 ˜  (p − 1)/(p − 1) if S = Np(k− )+ and b = 0,  2 2  R ˜ Iso (Nk,B; S; p) = pk−1(p − )b−1 S˜ = N b = ,  1 if p(k−2)+b(p−1)+2 and 0   0 otherwise.

7. Further remarks

The enumeration of graph coverings satisfying other properties has also been studied. In particular, Hofmeister [10], [12] introduced the notion of a concrete covering of a graph G and gave formulas for enumerating the concrete coverings of G. Later, Feng, Kim, Kwak and Lee [3] showed that the number of r-fold concrete coverings of G is equal to that of r-fold coverings of the join of G and an extra vertex. As a consequence, we can enumerate the concrete coverings of a 9 Enumerating coverings 197 graph by using known formulas for enumerating the coverings of a graph. This also gives a new formula for computing the number of graphs with n vertices, because the number of concrete double coverings of the complete graph on n vertices is equal to the number of graphs with n vertices. Corresponding results for regular coverings have also been obtained. For the enumeration of bipartite coverings, we refer to Archdeacon et al. [2], Hong et al. [17] and Kwak et al. [25]. In general, we may wish to enumerate the coverings of a graph whose covering graphs satisfy a given property P; for example, see [4] when P is a kind of Cayley graph. For surface coverings, the orientable coverings of a non-orientable surface of certain types (such as branched, unbranched, regular or irregular coverings) have been enumerated: in particular, the enumeration formulas for the orientable coverings of a non-orientable surface corresponding to Theorems 4.2, 4.3, 5.2 and 5.3 can be found in [6], [25] and [5]. As we mentioned, the number Iso (Nk,B; S; r) is not yet completely determined. When the folding number r is odd, this can be determined if we can count the connected branched coverings of a non-orientable surface with arbitrarily prescribed ramiﬁcation type.

References

1. J. W. Alexander, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920), 370–372. 2. D. Archdeacon, J. H. Kwak, J. Lee and M. Y. Sohn, Bipartite covering graphs, Discrete Math. 214 (2000), 51–63. 3. R. Feng, J. H. Kwak, J. Kim and J. Lee, Isomorphism classes of concrete graph coverings, 11 (1998), 265–272. 4. R. Feng, J. H. Kwak and Y.S. Kwon, Enumerating typical circulant covering projections onto a circulant graph, SIAM J. Discrete Math. 19 (2005), 196–207; 21 (2007), 548–550 (erratum). 5. I. P. Goulden, J. H. Kwak and J. Lee, Distribution of regular branched surface coverings, European J. Combin. 25 (2004), 437–455. 6. I. P.Goulden, J. H. Kwak and J. Lee, Enumerating branched orientable surface coverings over a non-orientable surface, Discrete Math. 303 (2005), 42–55. 7. J. L. Gross and T. W. Tucker, Generating all graph coverings by permutation voltage assignments, Discrete Math. 18 (1977), 273–283. 8. J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, 1987, and Dover, 2001. 9. M. Hall, Jr., Subgroups of finite index in free groups, Canad. J. Math. 1 (1949), 187–190. 10. M. Hofmeister, Concrete graph covering projections, Ars Combin. 32 (1991), 121–127. 11. M. Hofmeister, Graph covering projections arising from finite vector spaces over finite fields, Discrete Math. 143 (1995), 87–97. 12. M. Hofmeister, Enumeration of concrete regular covering projections, SIAM J. Discrete Math. 8 (1995), 51–61. 13. M. Hofmeister, A note on counting connected graph covering projections, SIAM J. Discrete Math. 11 (1998), 286–292. 14. A. Hurwitz, Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), 1–61. 198 Jin Ho Kwak and Jaeun Lee

15. A. Hurwitz, Über die Anzahl der Riemann’sche Flächen mit gegebenen Verzwei- gungspunkten, Math. Ann. 55 (1902), 53–66. 16. S. Hong, J. H. Kwak and J. Lee, Regular graph coverings whose covering transformation groups have the isomorphism extension property, Discrete Math. 148 (1996), 85–105. 17. S. Hong, J. H. Kwak and J. Lee, Bipartite graph bundles with connected fibers, Bull. Austral. Math. Soc. 59 (1999), 153–161. 18. G. A. Jones, Enumeration of homomorphisms and surface-coverings, Quart. J. Math. Oxford (2) 46 (1995), 485–507. 19. G. A. Jones, Counting subgroups of non-Euclidean crystallographic groups, Math. Scand. 84 (1999), 23–39. 20. J. H. Kwak, J. Chun and J. Lee, Enumeration of regular graph coverings having finite Abelian covering transformation groups, SIAM J. Discrete Math. 11 (1998), 273–285. 21. J. H. Kwak, S. Kim and J. Lee, Distributions of regular branched prime-fold coverings of surfaces, Discrete Math. 156 (1996), 141–170. 22. J. H. Kwak and J. Lee, Isomorphism classes of graph bundles, Canad. J. Math. 42 (1990), 747–761. 23. J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Theory 23 (1996), 105–109. 24. J. H. Kwak and J. Lee, Distributions of branched Dp-coverings of surfaces, Discrete Math. 183 (1998), 193–212. 25. J. H. Kwak, J. Lee and Y. Shin, Balanced coverings of a signed graph and some regular branched orientable surface coverings over a non-orientable surface, Discrete Math. 275 (2004), 177–193. 26. J. H. Kwak, A. D. Mednykh and J. Lee, Enumerating surface branched coverings from unbranched ones, London Math. Soc. J. Comput. Math. 6 (2003), 89–109. 27. V. Liskovets, On the enumeration of subgroups of a free group, Dokl. Akad. Nauk BSSR 15 (1971), 6–9. 28. V. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Appl. Math. 52 (1998), 91–120. 29. J. Lee and J. Kim, Enumeration of the branched mZp-coverings of surfaces, European J. Combin. 22 (2001), 1125–1138. 30. W. S. Massey, A Basic Course in Algebraic Topology, Springer, 1991. 31. A. D. Mednykh, Determination of the number of nonequivalent coverings over a compact Riemann surface, Soviet Math. Dokl. 19 (1978), 318–320. 32. A. D. Mednykh, On unramified coverings of compact Riemann surfaces, Soviet Math. Dokl. 20 (1979), 85–88. 33. A. D. Mednykh, Hurwitz problem on the number of nonequivalent coverings of a compact Riemann surface, Siber. Math. J. 23 (1982), 415–420. 34. A. D. Mednykh, Nonequivalent coverings over Riemann surfaces with a prescribed ramification type, Siber. Math. J. 25 (1984), 606–625. 35. A. D. Mednykh, On the number of subgroups in the fundamental group of a closed surface, Comm. Algebra 16 (1988), 2137–2148. 36. A. D. Mednykh and G. G. Pozdnyakova, Number of nonequivalent coverings over a non-orientable compact surface, Siber. Math. J. 27 (1986), 99–106. 37. M. Skoviera,˘ A contribution to the theory of voltage graphs, Discrete Math. 61 (1986), 281–292. 38. V. D. Tonchev, Combinatorial Configurations: Designs, Codes, Graphs, Wiley, 1988. 10 Symmetric maps

JOZEF ŠIRÁNˇ and THOMAS W. TUCKER

1. Introduction 2. Representations of maps 3. Regular maps 4. Cayley maps 5. Regular Cayley maps 6. Edge-transitive maps 7. Maps and mathematics References

A map is a graph embedding in which the focus is on the vertex-edge-face incidence relation together with its morphisms. This chapter concentrates on maps that are highly symmetric: regular maps, Cayley maps, regular Cayley maps and edge-transitive maps. In each case, we show how to construct a map using only information about the automorphism group. As a consequence, the viewpoint and methods are almost entirely group-theoretic.

1. Introduction

Symmetric maps – graphs embedded on surfaces with a sufﬁcient ‘level of symmetry’ – have been extensively studied over the last 100 years. Their roots, however, go much deeper, to the Platonic solids of the ancient Greeks and (much later) to Kepler’s stellated polyhedra. The theme of this chapter is how maps can be treated as purely algebraic structures. This is not just an exercise in abstract nonsense: such a viewpoint has historical origins, not only in Coxeter and Moser [11] but also in Edmonds’ original rotation schemes [13] and Tutte’s ﬂags [42], both of which view maps as permutation groups. Moreover, this algebraicization has a number of important

199 200 Jozef Širánˇ and Thomas W. Tucker consequences. It allows the simple description and construction of complicated maps – a group presentation is a remarkably efficient data structure. This in turn makes it possible to list all symmetric maps of moderate size, using computer software for group-theoretic calculations. For example, Conder [5] has determined all regular maps of genus at most 100, and Orbanicˇ [34] has found all edge-transitive maps with at most 1000 edges; such lists have been extraordinarily helpful for detecting general phenomena. Algebraicization also discloses and explains technical universal properties of maps and these, in turn, lead to connections with classical geometry. Finally, it exposes the field to all the tools of group theory: recent breakthroughs in the classification of regular maps [10] depend heavily on deep results from finite group theory. The chapter begins with a section giving a purely group-theoretic setting for maps in orientable and non-orientable surfaces. The next four sections survey results on four types of highly symmetric maps: regular maps, Cayley maps, regular Cayley maps and edge-transitive maps. The last section gives a brief glimpse into some remarkable connections between maps and other areas of mathematics, especially Galois theory and Riemann surfaces.

2. Representing maps algebraically

We first consider oriented maps – namely, maps on orientable surfaces with specified orientation. To describe such a map, we need only a rotation system: a permutation ρ on the set D of directed edges, sometimes called darts or arcs. We view ρ as acting on D on the right. Each cycle of ρ lists the darts beginning at a given vertex in the cyclic order determined by the orientation of the surface. The dart-set D also carries a fixed-point-free involution λ reversing the directions of each edge, with each cycle of λ representing the two directions of an edge. If we do not want to restrict λ to be fixed-point free, we can (and will) allow semi-edges – that is, edges with only one endpoint and only one direction (as opposed to loops, which begin and end at the same vertex and have two directions). To trace out the oriented face of the map containing the directed edge d, we first reverse d and then rotate it; we repeat the process until we return to d. Since we view permutations as acting on the right, the oriented faces for the map are given by the cycles of λρ. Thus, an oriented map M is a triple (D; ρ,λ), where ρ and λ are permutations of D and λ is an involution. The vertices of the map are the cycles of ρ, the edges are the cycles of λ, and the faces are cycles of λρ; vertex-edge, vertex-face and edge-face incidence are given by non-empty intersection. The permutation group on D generated by ρ and λ is called the dart group or monodromy group of the map, and is denoted by Mon(M). Since all our maps are assumed to be connected, Mon(M) is always transitive. Note that the monodromy 10 Symmetric maps 201 group does not determine the map; we must also specify the generators ρ and λ. For example, given the map (D; ρ,λ), its dual map is (D; λρ , λ), and its oppositely oriented map M− is (D; ρ−1,λ). It is important to view ρ and λ as giving instructions for assembling the map, not as motions or symmetries of the map. Rather, an (orientation-preserving) automorphism of the map (D; ρ,λ) is a bijection f : D → D that respects the right action of the dart group: f(dρ) = f(d)ρ and f (λ(d)) = f(d)λ, for all d ∈ D. (Transitivity guarantees that f is a surjection, so for finite maps we need not specify that f is an injection.) This agrees with the usual definition of an automorphism as a graph isomorphism respecting the rotation. It also clearly makes sense, since the conditions guarantee that f takes orbits to orbits and hence preserves all incidence relations. If we require that f(dρ)= f(d)ρ−1 instead, then we obtain an orientation-reversing automorphism.The collection of all orientation- preserving automorphisms of a map M forms a group denoted by Aut+(M). The collection of all automorphisms is denoted by Aut(M) and contains Aut+(M) as a subgroup of index at most 2. The action of Aut(M) on the map is viewed on the left, as usual. Automorphisms of a map generalize to branched coverings from one map to another. Given oriented maps M = (D; ρ,λ) and M = (D; ρ,λ),amorphism is given by f : D → D such that, for all d ∈ D,

f(dρ)= f(d)ρ and f(dλ)= f(d)λ.

Since again f takes orbits to orbits, it preserves incidence. Notice that f wraps faces (cycles of ρλ) around faces, also vertex neighbourhoods (cycles of ρ) around vertex neighbourhoods, and possibly takes edges to semi-edges. This means that we can view f topologically as a branched covering from the surface for M to the surface for M – namely, a local homeomorphism except at a finite number of ‘branch’ points at centres of faces, at vertices, and at midpoints of edges. If we want there to be no branch points at vertices or midpoints of edges (so that f can be viewed as an unbranched covering on the underlying graphs), then we must require that the restriction of f to any orbit of ρ or λ be injective. Algebraic connections can be carried even further by using standard methods from the theory of transitive permutation groups. Given a fixed d ∈ D, let d be the subgroup of = Mon(M) whose elements fix d, called the stabilizer of d in G. Then the dart set D can be identified with the set of right cosets d g, where g ∈ , so that the action of on these cosets by right multiplication is the same as the action of on D. Note that since is viewed as a set of permutations of D, it acts faithfully on D – that is, if dg = d for all d ∈ D, then g is the identity. − If h ∈ d and ghg 1 = h ∈ d , then d gh = d h g = d g,soh fixes the coset d g. It follows that Core(d ), the largest subgroup of d normal in , must be 202 Jozef Širánˇ and Thomas W. Tucker trivial when is a permutation group on D – that is, is core-free. For complete generality, we do not require d to be core-free. This means that if we want to think of Mon(M) as a permutation group on D, we have to replace by /Core(d ) and the right cosets of d by the right cosets of d /Core(d ) in /Core(d ). Automorphisms now have a simple algebraic interpretation. It is easy to see that if g normalizes d , then left multiplication by g also permutes the right cosets of d and respects the right action of on the right cosets of d .Itis a standard result that all automorphisms of the right action of on the right cosets + ∼ of d are obtained in this way, so that Aut (M) = N(d )/ d , where N(d ) is the normalizer of d in . Finally, it is not hard to show that morphisms of maps correspond to homomorphisms of dart groups, respecting dart stabilizers and choice of generators. We can summarize the algebraic view of oriented maps as follows:

Theorem 2.1 An oriented map M is given by a quadruple (, d ; ρ,λ), where =ρ,λ: λ2 = 1,... and d is a subgroup of .

(a)(Vertices, edges, faces) The right cosets / d of d correspond to directed edges, and the orbits of λ, ρ and λρ under right multiplication of on / d correspond to the edges, vertices and faces of the map M, with incidence given by non-empty intersection. (b)(Monodromy) As a permutation group, Mon(M) is /Core(d ) acting on the right cosets of d /Core(d ). (c)(Morphisms) A morphism from a map M = (, d ; ρ,λ) to a map M = ( ,d ; ρ ,λ) is a group homomorphism f : → such that f(d ) ⊆ d , f(ρ) = ρ and f (λ) = λ (since =ρ,λ, f must be surjective).A map morphism induces a branched covering with branch points, possibly at centres of faces, vertices and midpoints of edges. (d)(Automorphisms) The set of all automorphisms Aut+(M) is a group isomorphic to N(d )/ d , viewed as acting by left multiplication on / d .A morphism f satisfying ker(f ) ⊆ N(d ), corresponds to a regular branched covering and can be viewed as taking a quotient of M by a subgroup of Aut+(M). ∗ (e) (Associated maps) The dual of M = (, d ; ρ,λ)is M = (, d ; λρ , λ) and − − the opposite oriented map is given by M = (, d ; ρ 1,λ). An orientation- reversing automorphism of M is a morphism M → M−.

An orientable map M is regular if Aut+(M) acts transitively on the set of directed edges; regular maps are discussed in detail in the following section. In the algebraic context, M is regular if N(d ) = , so that d is normal in . The algebraic viewpoint leads to universal covering properties: 10 Symmetric maps 203

Theorem 2.2 Let M = (, d ; ρ,λ). Then the identity homomorphism f : → gives a covering from M to the trivial map (, ; ρ,λ) (a single semi-edge in the sphere) and a regular covering from the regular map M = (, {1}; ρ,λ) to M. If M has no semi-edges, all vertices have the same degree, and all faces have the same size, then the regular covering from M to M is unbranched.

Finally, the algebraic viewpoint provides a way of introducing geometry to maps. Let T (m, n) be the group x,y: xm = (xy)2 = (xy)n = 1.If1/m+1/n > 1 (T (m, n), { }; x,y) 2 , then the regular map 1 is one of the ﬁve Platonic solids viewed as tessellations of the sphere by n-gons, with m of them meeting at each vertex. If /m+ /n = 1 U(m, n) 1 1 2 , we get the standard tessellations of the Euclidean plane /m + /n < 1 by equilateral triangles, squares or regular hexagons. If 1 1 2 ,weget tessellations U(m, n) of the hyperbolic plane by regular n-gons, with m of them meeting at each vertex. The group T (m, n) is called a (2,m,n)-triangle group because it is isomorphic to the spherical, Euclidean or hyperbolic isometry group generated by rotations at vertices of a (π/2,π/m,π/n)triangle through angles 2π/2, 2π/m and 2π/n;a pair of adjacent triangles form a ‘fundamental domain’for the action of T (m, n) on U(m, n). The algebraic viewpoint again leads trivially to the following geometrical universal covering observation:

Theorem 2.3 Let M = (, d ; ρ,λ), where ρ and λ have orders m and n, respectively. Then the natural epimorphism f : T (m, n) → with f(x) = ρ and f(y)= λ gives a regular covering of M by a regular tessellation U(m, n) of the sphere, Euclidean plane or hyperbolic plane.

Jones and Singerman [26] chose this proposition as their foundation for a theory of maps. The viewpoint of permutation groups has been developed separately by various authors, in particular by Richter et al. [35]. For non-orientable maps, there are two basic approaches. One is that of generalized rotation systems in which edges are assigned the label 1 (untwisted) or −1 (twisted). The same embedding can be represented in different ways, since we can reverse the rotation at a vertex if we compensate by reversing the twisting of all edges incident to that vertex. Thus one must define an equivalence relation on generalized rotation systems based on such switchings. This approach is intuitive and topological, but not algebraic, and depends heavily on the structure of the underlying graph. The second approach is via flags – that is, vertex-edge-face incidence triples, viewed as triangles with sides from vertex to edge-midpoint, from vertex to face- centre, and from edge-midpoint to face-centre. In effect, this amounts to giving the map a barycentric subdivision, a viewpoint first suggested by Tutte [42]. Just 204 Jozef Širánˇ and Thomas W. Tucker as with oriented maps, we can form a monodromy group generated by three involutions:

τ, giving instructions on how to glue together vertex-edge sides of ﬂags; ω, giving instructions for edge-face sides (viewed as perpendicular bisectors of edges); θ, giving instructions for vertex-face sides (viewed as angle bisectors of a face corner).

Note that τωis an involution since four flag corners come together at the mid-point of each edge. Then, as before, viewing the action of = Mon(M) on the set of flags as the action of on the right cosets / a of the stabilizer a of a flag a, we have the following results:

Theorem 2.4 Consider the map M as a quintuple (, a; τ,θ,ω), where = τ,θ,ω: τ 2 = θ 2 = ω2 = (τω)2 = 1,....

• (Flags, vertices, edges, faces) The right cosets / a of a are the ﬂags, while the orbits of the subgroups ω,τ, τ,θ and ω,θ, under the action of on / a, are the edges, vertices and faces, respectively. • (Monodromy) As a permutation group, Mon(M) is /Core(a), acting on the right cosets of a/Core(a). • (Boundary) The singleton orbits of τ,θ and ω correspond to boundary components of the map; to eliminate such possibilities, one must require that no conjugate of these elements lies in a. • (Orientability) The subgroup ωτ, τθ has index 1 or 2 in . If there are no boundary components, then the subgroup has index 2 if and only if the map is orientable. When M is orientable, ωτ and τθ play the roles of λ and ρ for the oriented map. • (Morphisms) A morphism from (, a; τ,θ,ω) to ( ,a; τ ,θ ,ω) is a homomorphism f : → taking a into a and τ,θ,ω to τ ,θ ,ω, respectively. Morphisms correspond to branched-and-folded coverings. • (Automorphisms) The collection Aut(M) of all automorphisms of M forms a group isomorphic to N(a)/ a. • (Associated maps) Up to six different maps can be obtained from (, a; τ,θ,ω), depending on which two involutions one speciﬁes from τ, ω and τω to play the roles of ω and τ. • (Orientable double covering) If M is non-orientable, then the map Mo = ( × + o ∼ Z2,a ×{0}; (τ, 1), (θ, 1), (ω, 1)) is orientable with Aut (M ) = Aut(M) o + o and Aut(M ) = Z2 × Aut (M ). The projection from × Z2 to gives an unbranched regular covering from Mo to M. 10 Symmetric maps 205

Viewing maps as quotients of spherical, Euclidean or hyperbolic tessellations, we proceed as before, using the extended triangle group T ∗(m, n) =x,y,z: x2 = y2 = z2 = (xy)2 = (yz)m = (xz)n = 1, which is the isometry group generated by reﬂections in the sides of a π/2,π/m,π/ntriangle. This approach to general maps has been developed, for instance, by Bryant and Singerman [4] and, more recently, by Orbanicˇ [34]. We have not addressed the lifting of automorphisms for branched coverings or general constructions like Wilson’s parallel product [44]. Orbanic’sˇ thesis [34] provides a general setting, called an F -map, which is especially well suited for products and lifting, as well as the ‘hexagon’of dual and Petrie-dual maps of the penultimate part of Theorem 2.4.

3. Regular maps

Regular maps are the generalizations of the Platonic solids to surfaces of higher genus. The automorphism group of a Platonic solid acts transitively on vertices and on faces, so every vertex and face looks the same. The standard embedding of C3 C5 in the torus, obtained by identifying the sides of a 3 × 5 checkerboard, is vertex-transitive and face-transitive as a map, but it lacks the rotational symmetry around vertices and faces that a Platonic solid has. Thus, we call the map M regular if each vertex and face has such rotational symmetry – namely, an automorphism fixing the vertex or face-centre and rotating the incident edges one notch. Such rotational symmetry also ensures that M is vertex-transitive and face-transitive. In particular, all vertices have the same degree m and all faces have the same size n, making M an {m, n}-map. If v,e,f is a vertex-edge-face incidence in a regular map M and x and y are one-notch rotations about v and the centre of f , then xy reverses the direction of e. It follows that if M is regular and orientable, then Aut+(M) acts transitively (and hence regularly) on directed edges. Clearly, the converse also holds. Of course, the Platonic solids also have reflective symmetry: for each edge e there is an automorphism fixing the endpoints (and hence fixing the edge- directions) but interchanging the faces on either side of the edge. Call such an automorphism a reflection across e, and call a regular map reflexible if it has a reflection across some edge. Suppose M is a regular non-orientable map and C is an orientation-reversing cycle. Then beginning at a vertex v on C and performing one- notch rotations at vertices in order around C brings v back to v, but with orientation reversed. Since v has full rotational symmetry, it follows that v has full dihedral symmetry, so each edge incident to v has a reflection. Thus M is automatically reflexible. In a reflexible regular {m, n}-map, stabilizers under Aut(M) of vertices, faces and edges are, respectively, the dihedral groups Dm, Dn and D2 = Z2 × Z2. From the perspective of flags, a map M is reflexibly regular if and only if Aut(M) acts transitively (and hence regularly) on flags. 206 Jozef Širánˇ and Thomas W. Tucker

If M is orientable, it may not have any reflections; if so, call it chiral.We will show later that many regular maps are chiral. For a chiral regular {m, n}-map, Aut+(M) = Aut(M) and the stabilizers under Aut(M) of vertices, faces and edges are the cyclic groups Zm, Zn and Z2. From the previous section, we can identify an orientable regular map with its automorphism group Aut+(M). Thus, the study of orientable regular maps is the study of finite groups x,y: (xy)2 = 1,..., where vertices, faces and edges are cosets of x, y and xy, respectively. The associated map is reflexible if and only if the group has an automorphism f such that f(x)= x−1 and f(y)= y−1. Similarly, the study of reflexible maps is the study of groups r, s, t: r2 = s2 = t2 = (rt)2 = 1,..., where vertices, edges and faces are cosets of r, s, s,t and r, t, respectively. The map is orientable if and only if the subgroup rs,st has index 2. The classification of regular maps is therefore equivalent to the classification of quotients of triangle groups or extended triangle groups. This undoubtedly difficult task has been approached in three main directions: classification by automorphism group, by underlying graph, and by supporting surface. We discuss briefly each case.

Classiﬁcation by automorphism group A complete classiﬁcation of orientable regular and regular maps with a given orientation-preserving automorphism group is easy to obtain for cyclic and dihedral groups (see, for example, [30]). The problem becomes very hard for non-trivial classes of groups, however, and results are scarce. For example, it is known that any non-cyclic simple group is generated by two elements whose product is an involution (see [31]), so any such group is the orientation-preserving automorphism group of some regular map. On the other hand, a complete list of the possible maps for such a group is known only for the groups PSL(2,q) (see [36]). For other classes of simple groups we only have partial results by way of enumeration of regular {3, 7}-maps on the Ree groups (see [21]) and regular {4, 5}-maps on the Suzuki groups (see [24]).

Classiﬁcation by underlying graph In this direction we have, in theory, a complete characterization. Let G be a connected regular graph of degree d ≥ 3. By the results of Gardiner et al. [14], the graph G has an orientable regular embedding if and only if Aut(G) contains a subgroup A acting regularly on darts of G, such that the stabilizer of a vertex is a cyclic group of order d that acts regularly on darts at the vertex. For a corresponding statement for regular maps in general, we have to involve edge stabilizers as follows (see [14]): the graph G has a reﬂexible regular embedding if and only if Aut(G) contains a subgroup A acting transitively on darts, such that the stabilizer of a vertex is a dihedral group of order 2d whose cyclic part acts regularly on darts at the vertex, and such that the stabilizer of an 10 Symmetric maps 207

edge is isomorphic to Z2 × Z2. These characterizations are special cases of [38], where the underlying graphs of any vertex-symmetric map are characterized. The assumption on the edge stabilizer cannot be dropped, since there exist graphs fulfilling all the other assumptions. One example is the famous Coxeter graph, the unique cubic graph of girth 7 having 28 vertices. It is known that its automorphism group is isomorphic to P GL(2, 7). The only subgroup of P GL(2, 7) that has dihedral vertex stabilizers of order 2d = 6isPSL(2, 7). However, it can be checked that edge stabilizers in this group are isomorphic to Z4. So, despite its high degree of symmetry, the Coxeter graph does not admit a regular embedding. Among the most obvious concrete classes of symmetrical graphs, a complete characterization of the corresponding orientably regular or regular maps with a given underlying graph is known only for complete and complete bipartite graphs. (For the state-of-the-art regarding other classes of graphs, see Nedela [33].) Biggs [2] showed that orientably regular embeddings of Kn exist if and only if n is a prime power pe: the constructive part of his result uses Cayley maps (see the next section) derived from finite fields of order pe. James and Jones [20] proved that Biggs’s maps are the only examples of regular embeddings of Kn and that there are ϕ(n − 1)/e isomorphism classes of such maps, each corresponding to a conjugacy class of primitive elements of the finite field of order n = pe, where ϕ is the Euler function. The non-orientable counterpart was considered by Wilson [43], who showed that a regular non-orientable map with underlying graph Kn exists only if n = 3, 4 or 6; it is unique if n = 3 or 4, and there are two maps if n = 6. A classification of orientable regular embeddings of Kn,n follows from a deep analysis by Jones [23], in combination with the two earlier contributions [25] and [12]. It is folklore that there is an orientable regular embedding of Kn,n, for any n; in [23] there is a complicated formula for determining the number of all non- isomorphic orientable regular embeddings of Kn,n. As an interesting aside, this number is 1 if and only if there is a unique group of order n – that is, if and only if n is relatively prime to ϕ(n). Kwak and Kwon [28] have shown that a non-orientable regular embedding of Kn,n exists if and only if n ≡ 2 (mod 4) and every odd prime in the prime factorization of n is congruent to ±1 (mod 8). The number of such distinct regular embeddings is then 2k, where k is the number of distinct odd prime factors of n.

Classification by supporting surface Historically, this is the oldest direction. Besides trivial examples such as semi-stars (1-vertex maps with only semi-edges), cycles and dipoles, there are just five non-trivial orientably regular maps on a sphere – the maps of the five Platonic solids. Similarly, for the projective plane, besides cycles and dipoles there are only four non-trivial regular maps: 208 Jozef Širánˇ and Thomas W. Tucker

the regular embedding of K4 and its dual (a 3-cycle with doubled edges), and the regular embedding of K6 and its dual (the Petersen graph). Toroidal regular maps were classified by Coxeter and Moser long ago (see [11]). They fall into three classes according to their types ({3, 6}, {6, 3} and {4, 4}), and in each of them there are infinitely many chiral and reflexible maps. On the other hand, there is no regular map on the Klein bottle. The problem becomes very difficult for surfaces of negative Euler characteristic. By the well-known Hurwitz bound (see Chapter 11), if M is a map on the surface of Euler characteristic ε<0, then |Aut(M)|≤−84ε. It follows that each surface of negative Euler characteristic supports only a finite number of regular maps. By the late 1970s, regular maps had been classified on all orientable surfaces of genus at most 7, and on all non-orientable surfaces of crosscap number at most 8 (see [37] for references). A computer-aided classification, based on an adaptation of the low-index subgroup algorithm, was presented by Conder and Dobcsányi [6] for orientable surfaces of genus at most 15 and non-orientable surfaces of crosscap number at most 31. The most recent lists of Conder [5] go up to genus 100 and crosscap number 200. Until recently, however, there was no classification for infinitely many genera. The first breakthrough was a complete classification of regular maps on surfaces of negative prime Euler characteristic, obtained by Breda, Nedela and Širánˇ [3].

Theorem 3.1 Let p be an odd prime, p = 7, 13, and for p ≡−1 (mod 4) let ν(p) denote the number of pairs of odd positive integers (j, l) such that j>l≥ 3 and (j − 1)(l − 1) = p + 1. Then, up to isomorphism and duality, the number of regular maps of genus p + 2 is

0if p ≡ 1 (mod 12), 1if p ≡ 5 (mod 12), ν(p) if p ≡ 7 (mod 12), ν(p) + 1if p ≡ 11 (mod 12).

What is most striking about the situation for regular non-orientable maps are the gaps (that is, the absence of regular maps) for p ≡ 1 (mod 12) and p = 13. (In this connection, it is interesting to note that Conder and Everitt [7] proved that more than three-quarters of all non-orientable surfaces support at least one regular map.) By contrast, for orientable maps there is a regular map of every genus: the 2n regular map (Z4n, {1}; ρ,ρ ), where ρ generates Z4n, has 1 vertex, 2n edges and 1 face (since ρ2n+1 has order 4n), and hence has genus n − 1. This map is highly ‘degenerate’ in that it has multiple edges both in the primal and dual graph. It is also reﬂexible. 10 Symmetric maps 209

Therefore, it is reasonable to ask whether there are gaps for non-degenerate regular maps or for chiral regular maps. Computer lists (see [6], [5]) have shown that there are. Those lists provided the motivation and guidance for the following theorem of Conder, Širánˇ and Tucker [10] classifying all orientable regular maps of genus h for which |Aut+(M)| and h − 1 are relatively prime. Theorem 3.2 Let M be a regular orientable {k,m} map of genus h = 1. Let Aut+(M) = A =x,y: xk = ym = (xy)2 = 1,..., where the ellipsis indicates possible additional relators. Suppose that |A| and h−1 are relatively prime. Then there are the following possibilities. (a) Suppose that neither M nor its dual M∗ has multiple edges: then M is one of the five Platonic solids (so h = 0) and {k,m} is {3, 3}, {3, 4} or {3, 5} with no additional relators. (b) Suppose that M∗ has multiple edges but M does not, and that h>1; then either k = 4 and m = 3r(r>1) with the additional relator xy3x−1y3 = 1 or k = 3 and m = 8 with the additional relator xy4x−1y4 = 1 (the second case is a single map with h = 2). (c) If M and M∗ both have multiple edges and x∩y={1}, then either k = 2s and m = 2t(s>1,t > 1), gcd(s, t) = 1 with additional relators xy2x−1y2 = yx2y−1x2 = 1,ork = 8 and m = 3r(r>1) with additional relators xy3x−1y3 = yx4y−1x4 = 1. (d) If x∩y=C ={1}, then both M and M∗ have multiple edges and C is central with quotient one of the groups appearing in parts (a) – (c) (but allowing s,t = 1 in part (c)). The additional relator has the form xi = yj . In all cases, M is reflexible. We observe at the outset that M has multiple edges if and only if Core(x) is non- trivial (and similarly for M∗). We also observe that from elementary arithmetic with the Euler characteristic, the condition gcd(|A|,g− 1) = 1 ensures that A is ‘almost Sylow-cyclic’: its Sylow p-subgroups are cyclic if p is odd, and cyclic or dihedral if p = 2. The proof of Theorem 3.2 for parts (a) – (c) falls into two cases depending on whether A is solvable or not. If A is solvable then, since it is almost Sylow-cyclic, a minimal normal subgroup N p 2 N must be either Zp for odd prime ,orZ2 or Z2. Suppose that is cyclic. If it meets x or y non-trivially, then M or M∗ has multiple edges and, if not, then the regular covering M → M/N is unbranched, so |N| divides both |A| and g −1. N = 2 If instead Z2, we get either the tetrahedron for part (a) or we get parts (b) and (c). The non-solvable case involves the classification of almost Sylow-cyclic groups and leads to the cube–octahedron or the dodecahedron–icosahedron for part (a). The reflexibility of the maps follows easily from their presentations: each relation remains a relation when x is replaced by x−1 and y by y−1. 210 Jozef Širánˇ and Thomas W. Tucker

Theorem 3.3 Let M be an orientably regular map on the surface of genus p + 1, where p>13 is prime. Then p ≡ 1 (mod 6, 8 or 10), and Aut+(M) is the semi- direct product of Z2p with Z6, Zp with Z8,orZp with Z10, respectively. In all cases, M is chiral and has no multiple edges. In addition, M∗ has no multiple edges if p ≡ 1 (mod 8 or 10).

The three classes of maps in Theorem 3.3 were ﬁrst studied by Belilopetsky and Jones [1]. They are all balanced regular Cayley maps for cyclic groups, discussed later in this chapter. The proof of Theorem 3.3, which is not long, depends on observing that, for p>48, the Sylow p-subgroup of A is normal and its quotient is the automorphism group of one of the regular maps in the double torus S2, which are few and have already been classiﬁed.

Corollary 3.4 There is no chiral regular map on the surface Sp+1, when p is prime and p ≡ 1 (mod 6, 8 or 10).

Corollary 3.5 Any regular map M of genus p + 1 has multiple edges either in M or in M∗, when p is prime and p ≡ 1 (mod 8 or 10).

Regular maps of genus p + 1, where p is prime, are completely classiﬁed by Theorems 3.2 and 3.3. The full classiﬁcation is rather complicated, because of the different types of presentations and the number of parameters needed to specify the orders of x and y and the relation xi = yj . In addition, Theorem 3.1 classifying regular non-orientable maps of Euler characteristic −p follows directly from Theorems 3.2 and 3.3 using orientable double coverings.

4. Cayley maps

The labelling of edges of a Cayley graph C(A,X) by the generating set X affords us the opportunity to give an orientable embedding, simply by giving a cyclic order ρ for the set X consisting of the elements of X and their inverses. Such an embedding is called a Cayley map for A and is denoted by CM(A,X,ρ).In particular, the regular action of A on the Cayley graph C(A,X) given by left- multiplication preserves edge labels, so it preserves the rotation and extends to a regular action by A on the Cayley map CM(A,X,ρ), making it a strongly symmetric embedding of C(A,X) in the language of Chapter 11. Conversely, any orientable map whose automorphism contains a subgroup A acting regularly on the vertices is a Cayley map for A and some X and ρ.

Theorem 4.1 Let M be an orientable map and let A be a group. Then the following statements are equivalent: 10 Symmetric maps 211

• M is a Cayley map for A; • A is a subgroup of Aut+(M) acting regularly on the vertex-set; • M is a strongly symmetric embedding of a Cayley graph for A; • M is the derived embedding of a 1-vertex embedded voltage graph whose voltages generate A.

The distribution of inverses for a Cayley map CM(A,X,ρ)is the involution τ deﬁned on X by τ(x) = x−1, for all x ∈ X; the elements of X left ﬁxed by τ are the involutions in X. The faces of the Cayley map are determined by τ in the following sense: if w is the product of the elements of X in a cycle of ρτ, and if r is the order of w in A, then there are |A|/r faces in the Cayley map, each of whose boundaries is given by wr . This can be seen simply by tracing out faces using ρ and τ, or by noting that the cycles of ρτ give the faces in the 1-vertex embedded voltage graph. Given a Cayley map CM(A,X,ρ) a natural question to ask is how automorphisms of the map are related to group automorphisms of A.

Example 1 Let A be the dihedral group x,y: x2 = y4 = (xy)2 = 1. Then the Cayley map with rotation (x,y,y−1) is the cube, but the map automorphism that rotates through 120◦ about the identity vertex is not a group automorphism, since it takes the involution x to the non-involution y. On the other hand, the reﬂection along the edge x taking y to y−1 is a group automorphism. If a map automorphism ﬁxing the identity vertex is to be a group automorphism, then f(x−1) = f(x)−1 for all x ∈ X – that is, the permutation of X induced by f commutes with τ. The converse is also true:

Theorem 4.2 Let f be an automorphism of the Cayley map CM(A,X,ρ) that leaves the identity vertex ﬁxed. Then f gives a group automorphism of A if and only if the permutation of X induced by f commutes with τ.

The proof ﬁrst shows that f(xy)= f(x)f(y)for all x,y∈X, with xy viewed as a path of length 2 beginning at the identity vertex, whose ‘angle’ is the same as the angle between τ(x)and y. Since f preserves angles (it respects the rotation ρ) and commutes with τ, the path f(x)f(y)has the same initial point and same angle, so f(x)f(y) = f(xy). By induction, the same argument extends to f(xa) = f(x)f(a)for any a ∈ A,sof is an automorphism. A Cayley map CM(A,X,ρ)is balanced if τ is a power of ρ. Since τ has order at most 2, the only possibilities are that the degree d is even and τ = ρd/2 (so x and x−1 are antipodal), or that τ is the identity (so all x ∈ X are involutions); the former is called type I and the latter type II (see [40]). Since any map automorphism f for a Cayley map satisﬁes fρ = ρ±1f , f commutes with τ if the map is balanced. Therefore: 212 Jozef Širánˇ and Thomas W. Tucker

Corollary 4.3 For a balanced Cayley map, any map automorphism leaving the identity fixed is a group automorphism. Example 2 Let GF (q) be the finite field of order q = pn > 4, for p prime, let ∼ n A = Zp be its additive group, and let x be a generator of its cyclic multiplicative group of order q − 1. Denote the additive identity by 0 and the multiplicative identity by 1. Let M = CM(A,(1,x,x2,...,xq−2)), whose underlying graph is the complete graph Kq . Multiplication by x is an automorphism of A and, as it clearly respects the rotation, is also a map automorphism of M. Since this automorphism rotates the map one notch around the vertex 0, the map M is regular. Since x(q−1)/2 =−1, it follows that M is a balanced Cayley map, of type I if p = 2 and type II if p = 2. If M were reflexible, then there would be a map automorphism of order 2 interchanging xi and xq−1−i = x−i, for all i. Since M is balanced, by Corollary 4.3 this map automorphism would also have to be an additive automorphism of A; but this is clearly impossible, since for q>4 there are always y,z ∈ GF (q) such that (y +z)−1 = y−1 +z−1. Thus the map is chiral. Example 2 has an interesting history. It was used by Edmonds to disprove a conjecture in the first edition of Coxeter and Moser [11] that there are no chiral regular maps of genus g>1. Although he never published the result, it inspired him to think of rotation schemes as a way of describing all oriented maps (see [13]). It was later rediscovered by Biggs [2] and, as discussed in the previous section, it gives us all those orientable regular maps whose underlying graph is complete. In particular, there are no regular orientable reflexible maps with complete underlying graph Km, for m>4. The same ideas apply to a regular balanced Cayley map M = CM(A,X,ρ) when A is cyclic. By Corollary 4.3, if M is reflexible, there is a group automorphism of A fixing x ∈ X; as long as the degree d>2, this group automorphism must be non-trivial. Since ρ also induces a group automorphism of A with the orbit of x generating A, it follows that x is a generator of A. But no non-trivial automorphism of a cyclic group can fix a generator. Corollary 4.4 Any balanced regular Cayley map of degree d>2 for a cyclic group is chiral.

5. Regular Cayley maps

Throughout this section, all surfaces are orientable and all automorphisms are orientation-preserving. A Cayley map M = CM(A, X) that is also regular has, in some sense, the most possible symmetry: Aut+(M) acts regularly on the directed edges and its subgroup A acts regularly on the vertices. The focus of this section is to determine 10 Symmetric maps 213 which groups A have a regular Cayley map CM(A, X); as we shall see, this is an extremely difﬁcult and complicated problem. In contrast, every group has many Cayley maps. Thus we should view regular Cayley maps as rare and unusual. It is therefore surprising that they behave as universal objects, in the following sense [35]:

Theorem 5.1 Let M1,M2,... ,Ms be any collection of finite maps and let d be the least common multiple of all the degrees. Then there is a d-valent regular Cayley map M which regularly covers each of the maps in the collection: there may be branch points at the vertices. The map M can be chosen to be finite or planar (infinite). Moreover, M can be chosen so that it is a Cayley map with respect to all possible distributions of inverses.

Let M = CM(A, X) be a regular Cayley map of degree d, and let y be the automorphism rotating one notch about the identity vertex. Then each element of Aut+(M) can be written uniquely as ayi – that is, Aut+(M) has a factorization Ay. Moreover, since M is regular, Aut+(M) =λ, y is generated by y, together with an involution λ. Conversely, any group λ, y that factors Ay for some subgroup A is the automorphism group of a regular Cayley map for A, since A acts regularly on the cosets of y. Thus we have the following result:

Theorem 5.2 The group y,λ: λ2 = 1,... is the automorphism group of a regular Cayley map for the group A if and only if it has a factorization Ay.

Example 3 The (orientation-preserving) automorphism groups of the tetrahedron, cube–octahedron and dodecahedron–icosahedron are, respectively, A4, S4 and A5. For the tetrahedron, A4 =(123), (12)(34)=y,λ. Moreover, the even ∼ involutions of A4 form a subgroup A = Z2 × Z2 and A4 = Ay; thus, the tetrahedron is a regular Cayley map for Z2 × Z2. The group S4 has two factorizations, one as D4(123) and the other as S3(1234). The first factorization shows that the cube is a regular Cayley map for D4, and the second shows that the octahedron is a regular Cayley map for S3 (for both maps, λ = (34)). Finally, A5 has a factorization as A4(12345); thus, the icosahedron is a regular Cayley map for A4 with λ = (12)(34). On the other hand, A5 has no factorization Ay with y of order 3, since the permutation action of A5 on the cosets of A would give a non-trivial homomorphism from A5 into S3, contradicting the simplicity of A5; thus, the dodecahedron is not a regular Cayley map. Suppose that the regular Cayley map M = CM(A, X) is balanced. Then the map automorphism rotating one notch around the identity vertex is also a group automorphism f of A, and X is the orbit under f of any element x ∈ X.IfM is of type I, then the degree d is even and f d/2(x) = x−1;ifM is type II, then x is an involution. An automorphism of the group A is balanced for x if the orbit of 214 Jozef Širánˇ and Thomas W. Tucker x under f generates A and contains the inverse of x. Thus any group A having a regular balanced Cayley map has a balanced automorphism. Conversely, if f is a balanced automorphism of A for x, then the Cayley map CM(A,X,(x,f(x),f 2(x), . . .)) is regular, since f respects the rotation and is therefore also a map automorphism. If x is not an involution and if f i(x) = x−1, then f 2i(x) = x, making f 2i the identity, so the degree is d = 2i and the map is balanced of type I; if x is not an involution, then all elements in the orbit of x are involutions, since f is a group automorphism, and the map is balanced of type II. In terms of the factorization Aut+(M) = Ay for a regular Cayley map, being balanced is equivalent to A being normal in Aut+(M), with conjugation by y providing the balanced automorphism. In summary we have: Theorem 5.3 The group A has a regular balanced Cayley map if and only if it has a balanced automorphism. The regular Cayley map M = CM(A,X,ρ) is balanced if and only if A is normal in G = Aut+(M). For example, of the Platonic solids, only the tetrahedron is a balanced regular Cayley map, since S4 has no normal subgroups of index 3 or 4 and A5 is simple. In some cases, finding a balanced automorphism is not hard: Theorem 5.4 Let A =x,y, where x is an involution and y has finite order. If the normal closure N(x) = A, then A has a balanced automorphism. In particular, the following all have balanced automorphisms: simple groups, the symmetric groups Sn, for n>2, and any group A =x,y where x is an involution and the orders of y and xy are relatively prime. Proof Since the conjugates of x by y generate Nx=A and x is an involution, conjugation by y is balanced for x. It is a consequence of the classification of finite simple groups that every simple group is generated by two elements, one of which is an involution. It is easy to find elements x and y generating Sn with x a single transposition, so that x is not contained in An, the only normal subgroup of Sn. Finally, since the images of both y and xy are separately generators of the cyclic group A/Nx, if their orders are relatively prime, then A/N is trivial. On the other hand, even for cyclic groups, the situation can be surprisingly complicated. The trivial Cayley map for Zn of degree 2 in the sphere is regular and balanced. But are there non-trivial examples? The answer is purely number- theoretic [8]: Theorem 5.5 Suppose that n = 2km, where m is odd. Then the cyclic group Zn fails to have a non-trivial balanced regular Cayley map if and only if either m = 1,k >1,orm is a product of distinct Fermat primes, one of which must be 3 if k = 0 or 1. 10 Symmetric maps 215

Proof We give the proof, since it is short and illustrates the role of elementary number theory in the study of regular Cayley maps for abelian groups. Any automorphism of Zn is given by multiplication by some w relatively prime to n. To get a non-trivial balanced automorphism, we require that wr ≡−1 (mod n), where w ≡−1 (mod n). By the Chinese Remainder Theorem, the multiplicative ∗ ∗ group Zn of residues relatively prime to n is the direct product of groups Zpe for e ∗ the unique factorization of m into prime powers p . If the order |Zn| has an odd ∗ factor, then Zn has an element w of odd order, so −w is a non-trivial root of −1. ∗ e−1 ∗ Since |Zpe |=p (p − 1), the only way that |Zn| can fail to have an odd factor is if m = 1 or if all odd primes p in the prime factorization of m have multiplicity 1 and have the form 2s + 1 – that is, m is a product of distinct Fermat primes. To avoid even-order roots of −1, we require that either 4 divides n or that one of the Fermat primes dividing m is 3, since there is no square root of −1 (mod 4) and 3 is the only Fermat prime that has square root −1.

Example 4 If p is an odd prime and if 2r divides (p − 1), then Zp has an element w of order 2r,sowr ≡−1 (mod p), since p is prime. The 1-vertex voltage graph for the associated balanced regular Cayley map has two faces, each with voltage

r− r 1 − w + w2 −···+w 1 = (w + 1)/(w + 1) ≡ 0 (mod p) when r is odd, or one face with voltage

r− r 1 − w + w2 −···−w2 1 = (w2 − 1)/(w + 1) ≡ 0 (mod p) when r is even. The resulting map has genus

h = − 1 (p − rp + p) = + 1 (r − )p r , 1 2 2 1 2 3 when is odd h = − 1 (p − rp + p) = + 1 (r − )p r . and 1 2 1 2 2 when is even

Of particular interest are the cases r = 5 and r = 4, where h − 1 = p: these are two of the Belilopetsky–Jones regular maps, discussed in the previous section. The third type of these maps corresponds to r = 3, but using the cyclic group Z2p instead, since otherwise the map is toroidal. All these maps are chiral, by Corollary 4.4. Determining which non-cyclic abelian groups have a balanced automorphism is far more complicated. If f is balanced for the non-involution x, then the semi- order is half the order of f . The basic approach for abelian groups is to write them as a direct product of their Sylow p-subgroups. We observe that if |A| and |B| are relatively prime, then any automorphism of A × B that is balanced for (a, b) must leave A and B invariant, and hence must induce balanced automorphisms for A and B for a and b, respectively. Deﬁne the 2-part of an integer m to be the highest 216 Jozef Širánˇ and Thomas W. Tucker power of 2 dividing m. It is easy to check that the semi-orders of the induced automorphism have the same 2-part. The converse is just as straightforward, so we have the following result. Theorem 5.6 Let A and B be groups that are not generated by involutions for which |A| and |B| are relatively prime. Then A×B has a balanced automorphism if and only if both A and B have balanced automorphisms with semi-orders having the same 2-part. This result is good news and bad news: it is good because it says that we can put together balanced automorphisms for the Sylow p-subgroups of an Abelian group, but bad because it says that we have to worry about the 2-parts of the semi-orders. Call a group 2k-good if it has a balanced automorphism whose semi-order has 2-part 2k; for example, by [8], every group A of odd order is 1-good. On the other hand, it is not hard to show that Z2 × Z4 is 2-good, but not 1-good. Thus, if A is Abelian and its Sylow 2-subgroup is Z2 × Z4, all the Sylow p-subgroups of A for odd primes p must be 2-good, in order for A to have a balanced automorphism. This is where the situation gets increasingly complicated. When p is an odd prime and k>1, determining whether an Abelian p-group A is 2k-good depends e e A 1 × 2 × on detailed information about the canonical form of as the product Zm1 Zm2 ..., where m1 |m2 | .... Call the exponents e1,e2,... the ledge numbers of A. Let A be an Abelian p-group. It is shown in [8] that A is 2-good when p ≡ 1 (mod 4), but when p ≡ 3 (mod 4) then A is 2-good if and only if all the ledge numbers for A are even. This is a special case of the following remarkable result (see [32]): Theorem 5.7 Let A be an Abelian p-group, where p is an odd prime. Then A is 2k-good if and only if all the ledge numbers are divisible by the multiplicative order of p in Z∗ . 2k+1 For Abelian 2-groups, the situation gets almost out of hand. It is easy to give examples such as Z2 × Z8 that have no balanced automorphisms. The full characterization was achieved by Muzyhchuk (see [32]); the conditions are highly e e 1 × 2 technical. Call an Abelian 2-group almost homocyclic if it has the form Zm Z2m. e An Abelian p-group is elementary if all elements have order p – that is, A = Zp. In addition, let ir(n) be the number of irreducible polynomials of degree n over the 2-element ﬁeld GF (2), where ir(1) = 1; for example, ir(2) = 1, since the only irreducible polynomial of degree 2 over GF (2) is x2 + x + 1. Theorem 5.8 The non-elementary Abelian 2-group A is 2k-good if and only if it n is the product of groups of the form An, where An is itself the product of at most k ir(n) groups Bn,i, where Bn,i is either almost homocyclic of rank 2 or elementary Abelian of rank at most 2k. As an example of what this theorem says, we give the following consequence: 10 Symmetric maps 217

Corollary 5.9 Let A be a non-cyclic Abelian 2-group, all of whose ledge numbers are 1. Then A has a balanced automorphism if and only if A = Zm × Z2m.

Theorems 5.7 and 5.8 together provide a way of determining whether anAbelian group A has a balanced automorphism. It is, however, astonishing that such a simple question about the existence of an automorphism of an Abelian group can have such a complicated answer. Moreover, it appears that the conditions of Theorem 5.8 cannot be verified in any computationally efficient way from, say, the canonical form of an Abelian 2-group. So far, we have considered only balanced regular Cayley maps. It is possible for a group to have a regular Cayley map, but not a balanced one. For example, although Z4n has no non-trivial balanced regular Cayley map, by Theorem 5.5 it does have a non-trivial regular Cayley map (which is ‘anti-balanced’, see [9]). This leaves open the case of cyclic groups of odd order. Indeed, [8] was motivated by the conjecture that Z15 has no non-trivial regular Cayley map, a fact first established by computer. The following characterization for all cyclic groups depends on the use of Ito’s theorem [18] that a group with factorization AB, where A and B are Abelian, has an Abelian commutator group:

Theorem 5.10 If n is the product of distinct Fermat primes, one of which is 3, then Zn has no non-trivial regular Cayley map.

For a general Abelian group A, suppose that M is a regular Cayley map for A and Aut+(M) =λ, y=AY . Let D = N(λ), let C be the commutator subgroup, and let B = A ∩ C. Then a detailed analysis of the relationship of these subgroups, using a generalization of Ito’s theorem, gives the following non- existence theorem [8].

Theorem 5.11 For each e, there exists an r such that no Abelian group with ledge number at most e and rank at least r has a regular Cayley map.

Non-balanced maps for general groups have been little studied. There are no results analogous to Theorem 5.6. The factorization Ay is also of little help, since such factorizations, although common, are notoriously difﬁcult to use: Ito’s theorem is the only general result and is not available when A is non-Abelian. Jajcay and Širánˇ [19] introduced the idea of a skew morphism f : A → A, deﬁned naturally for Ay by the equation ya = f(a)yi, where i = π(a) is a function of a, and satisfying f(ab) = f(a)fπ(a)(b). For example, a regular Cayley map is balanced if and only if π(a) = 1 for all a. Skew morphisms have been studied in a number of papers (see [19], [9], [35]) and allow exhaustive computer searches for small examples, but there is no general method for constructing or detecting them. 218 Jozef Širánˇ and Thomas W. Tucker

6. Edge-transitive maps

Suppose that M is an orientable map for which A = Aut(M) acts transitively on the edges. What sorts of symmetries might M have? We focus on the situation around a single edge e, since locally the map looks the same around every edge. First, we concentrate on the stabilizer Ae. Let u and v be the endpoints of e, and let f and g be its incident faces; we assume that neither M nor its dual has loops, so u = v and f = g. Since the only automorphism in Ae ﬁxing each of u, v, f and g is the identity, and since any automorphism takes vertices to vertices and faces to faces, Ae must be isomorphic to a subgroup of Z2 × Z2. We denote the three non-identity possibilities as follows:

τ: a transverse reflection fixing u and v, λ: a longitudinal reflection fixing f and g, φ: a 180◦ rotation interchanging u with v and f with g.

The notation here is that of Graver and Watkins; elsewhere in this chapter we think of λ not as a reﬂection, but as a 180◦ rotation, or as an element of Mon(M), which is not even an automorphism. At the vertices and faces incident to e we may ﬁnd any of the following automorphisms:

ρ: a one-notch rotation around u or v, σ : a one-notch rotation around the centre of f or g, θ: a corner reﬂection at any of the four face-corners deﬁned by the pairs (u,f ),(u,g),(v,f ),(v,g), γ : a glide taking e1 to e and e to e2, where e1 and e2 are the other edges at the corners (u, f ) and (v, g), respectively (or (u, g) and (v, f )).

We observe that ρ,σ and φ preserve orientation, while λ, τ, θ and γ do not. In addition, we have the relations φ = λτ, ρ = θτ and σ = θλ. A Petrie path in the map M is a path whose edges follow the zig-zag sequence e1,e,e2 that deﬁnes a glide. If such a path begins and ends at the same corner, we call it a Petrie cycle. Just as each edge appears once in two different face-cycles, or twice in the same face-cycle, each edge also appears once in two Petrie cycles or twice in one Petrie cycle. This means that we can create another map from M, the Petrie dual MP , whose face-cycles are the Petrie cycles. Petrie duality interchanges Petrie cycles and faces in the same way that usual duality interchanges vertices and faces. Another way to see the Petrie dual of a map is in terms of vertex-rotations and edge-twistings for the underlying graph: the Petrie dual is obtained simply by changing all twisted edges to untwisted and vice versa. It follows from this viewpoint that, when M is orientable, MP is orientable if and only if the underlying 10 Symmetric maps 219 graph is bipartite, and that Aut(MP ) and Aut(M) act in exactly the same way on the underlying graph of M. Finally, in terms of the algebraic representation of P maps of Section 2, we have (, a; τ,ω,θ) = (, a; τ,τω,θ). Example 2 If M is the cube, viewed as a map, it is not hard to check that all its Petrie cycles have length 6. This means the Petrie dual MP is a map in the torus, with the cube as its underlying graph. Note that MP is regular, since M is regular.

There are various ways that Ae can combine with the other kinds of automorphisms in an edge-transitive map. Graver and Watkins [15] determine there are fourteen types, summarized in Table 1. The names of the types and the notation for automorphisms are basically the same as theirs, although they denoted vertex-rotations by σu and face-rotations by σf . The second column gives the non- identity elements of Ae, the third column gives a canonical generating set for Ae, the fourth column gives all other symmetries (where, for example, ρ2 indicates the presence of a two-notch rotation, but not a one-notch rotation), and the ﬁfth column tells whether Ae acts transitively on vertices, faces or Petrie cycles. Note that duality interchanges τ with λ and ρ with σ, while Petrie duality interchanges τ with φ and σ with γ .

Table 1. Types of edge-transitive maps

Type Ae generators also present transitivity

1 τ,λ,φ τ,λ,θ all reﬂexible regular (all) 2 2 2 ττ,θ1,θ2 ρ,σ ,γ not vertex ∗ 2 2 2 λλ,θ1,θ2 ρ ,σ,γ not face P 2 2 2 φφ,θ1,θ2 ρ ,σ ,γ not Petrie 2ex τ τ, σ ρ2,γ2 all ∗ 2ex λλ,ρσ2,γ2 all P 2ex φφ,ρσ chiral regular (all) 2 2 2 3 θ1,θ2,θ3,θ4 σ ,ρ ,γ none 2 2 4 ρ,θ1,θ2 σ ,γ not vertex ∗ 2 2 4 σ, θ1,θ2 ρ ,γ not face P 2 2 4 γ,θ1,θ2 ρ ,σ not Petrie 2 2 5 ρ1,ρ2 σ ,γ not vertex ∗ 2 2 5 σ1,σ2 ρ ,γ not face P 2 2 5 γ1,γ2 ρ ,σ not Petrie

In addition, we know that the orientation-preserving automorphisms form a subgroup of index at most 2 in Aut(M). Thus, we have the following condition: Condition 1 The group A has a subgroup of index at most 2, containing all powers of φ,ρ and σ but not τ,λ,θ or γ . 220 Jozef Širánˇ and Thomas W. Tucker

Following Širánˇ et al. (see [39]) and the general theme of this chapter, we want to take the group information from Table 1 and construct a map of the given type. Suppose that the group A has one of the generating sets in the table, where τ,λ,φ and θ are involutions and τλ = φ. We can form a map where edges are left cosets of the edge stabilizer Ae, vertices are left cosets of the vertex stabilizers Au,Av, and faces are left cosets of the face stabilizers Af ,Ag; note that although the table does not give Au and Af , it is easy to ﬁnd generators for them, knowing that the possible elements of Au are τ,ρ,θ and the possible elements of Af are λ, σ, θ. We would like to say that the resulting map has the desired type as long as Condition 1 is satisﬁed, but there is a problem. Suppose that there is an automorphism of the group that acts in the way that conjugation by τ,λ,orφ does. It would also act on the stabilizers Ae,Au and Af in the same way, providing an extra symmetry for the map. It is not hard to show that the following condition bans such extra symmetry:

Condition 2 Apart from conjugation by an element of Ae, there is no involutory automorphism of A that ﬁxes all elements of Ae and permutes without ﬁxed points the other elements of X, where X is the generating set in the third column of Table 1.

A fundamental problem for edge-transitive maps is to determine which types are possible, in general or for a given surface. When Graver and Watkins [15] first described the different types, they were interested in the case of infinite planar maps and showed that only seven of the types are possible. It was also known (see [15]) that only three of the types are possible for finite planar maps. That all types are possible as finite maps follows from Širánˇ et al. (see [39]):

Theorem 6.1 For all n ≡ 11 (mod 12), and for any given type, there is an edge- transitive map with that type whose automorphism group is the symmetric group Sn.

The proof is simply a matter of giving generating sets for Sn of the desired type satisfying Conditions 1 and 2. For Condition 1, we require the permutations for τ,λ,θ and γ to be odd. For Condition 2, we use the fact that all automorphisms of Sn are inner automorphisms and that conjugate permutations have the same cycle structure. It is not known whether there is a single surface supporting all fourteen types. One approach is to use the existence of one type to derive another type in the same surface; for example, this can always be done with duality, so we need to ﬁnd only one from each of the dual pairs for 2, 2ex, 4 and 5. We can also use Petrie duality, although to maintain orientability we need the underlying graph to be bipartite. In this case, the underlying surface will probably change since the Petrie cycles do not generally have the same lengths as the face cycles. 10 Symmetric maps 221

Another well-known construction, however, does give maps in the same surface. Given a map M, its medial map Med(M) is obtained by truncating each vertex to form a 4-regular map, where the new vertices are the midpoints of the original edges, and where two new vertices are joined by an edge if the corresponding edges in the original map form a corner of a face: note that the underlying surface is still the same. Moreover, in some cases if M is edge-transitive, so is Med(M), but usually of a different type (see [39]):

Theorem 6.2 If M is edge-transitive of type 1 or 2exP , but is not self-dual, then Med(M) is edge-transitive of type 2∗ or 5∗, respectively. If M is edge-transitive of type 2ex or 2ex∗, then Med(M) is edge-transitive of type 4∗.

Thus, if we are looking for a surface supporting all fourteen types, by using duality and medial maps, we need only find a surface supporting the types 1, 2P , 2ex, 2exP , 3, 4P and 5P ; for example, for the sphere, the only (finite) edge- transitive maps are the five Platonic solids and their medial maps or duals, giving nine altogether of types 1, 2 and 2∗. For the plane (infinite), the possible types are 1, 2, 2∗, 2P , 3, 4, 4∗ and 4P (see [15]). Edge-transitive maps in the torus are completely classified in [39]: the possible types are 1, 2, 2∗, 2P , 2exP , 5 and 5∗. Since there are well-known toroidal maps of type 1 (hexagonal tilings) and type P P 2ex (the triangulation of the torus by K7), the only surprising type is 2 . Finally, we note that there are two kinds of face in Med(M), corresponding to the vertices and faces of the original map, so Med(M)∗ is bipartite and (Med(M)∗)P is orientable. Thus, if we are interested only in showing the existence of orientable maps of all types, not necessarily in the same surface, we only need one of type 3, since there are plenty of well-known examples of non-self-dual reflexible regular maps and bipartite chiral regular maps.

7. Maps and mathematics

There are fascinating connections, dating back to the 19th century, between maps in general (and regular maps in particular) and groups, hyperbolic geometry, Riemann surfaces and Galois theory. We now trace out the relationships between maps and some other branches of mathematics, restricting ourselves to oriented surfaces. As we observed in Section 2, each orientable map M can be viewed in two ways: as a branched covering over a trivial map M0 in the sphere consisting of one semi-edge, or as regularly branched covered by a tessellation U(m, n) of the sphere, Euclidean plane or hyperbolic plane. We can now introduce a complex structure on the supporting surface S for M (and hence turn S into a Riemann surface) by pulling down this structure from the hyperbolic plane by means of the 222 Jozef Širánˇ and Thomas W. Tucker covering U(m, n) → M, or by lifting it from the Riemann sphere to S with the help of (the inverse image of) the covering M → M0. Complex analysis gives us many other ways of making S into a Riemann surface. The remarkable feature of doing this with the help of maps, as sketched here, shows up at the level of viewing Riemann surfaces as complex algebraic curves. As Riemann showed, S can be described by a polynomial f(x,y) in two complex variables with complex coefficients, by way of evaluating y from the equation f(x,y) = 0 as a many-valued function of x and recovering S as a branched covering of the complex sphere. The coefficients of f may be arbitrary complex numbers. But Weyl’s rigidity theorem, combined with its converse proved by Bely˘ı (see Jones [22] or Lando and Zvonkin [29] for details), tell us that, loosely speaking, a compact Riemann surface S can be defined by a polynomial f(x,y) with algebraic coefficients if and only if S arises as a branched covering space by a suitable tessellation U(m, n), or equivalently, as a regular quotient of the complex upper half-plane by a suitable subgroup of a triangle group T (m, n), as described in Section 2. This correspondence between maps and Riemann surfaces can be used to study the absolute Galois group, the group of all automorphisms of the field of all algebraic numbers. The absolute Galois group is one of the most mysterious resistant groups appearing in any area of mathematics. Inspired by the Bely˘ı–Weyl theorem, Grothendieck proposed to study the absolute Galois group by extending its action on the (algebraic) coefficients of polynomials defining Riemann surfaces to the action on the corresponding maps. It can be shown that this action is faithful – even when restricted to embedded trees on a sphere – so this way of studying the absolute Galois group does not lead to loss of information. For more about these connections and for references see the excellent survey articles by Jones and Singerman [27] and Jones [22], or the monograph by Lando and Zvonkin [29].

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33. R. Nedela, Regular maps – combinatorial objects relating different fields of mathematics, J. Korean Math. Soc. 38 (2001), 1069–1105. 34. A. Orbanic,ˇ Ph.D. thesis, University of Ljubljana, 2006. 35. R. B. Richter, J. Širán,ˇ R. Jajcay, T. W. Tucker and M. E. Watkins, Cayley maps, J. Combin. Theory (B) 95 (2005), 189–245. 36. C. H. Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 13–42. 37. J. Širán,ˇ Regular maps on a given surface: a survey, Topics in Discrete Mathematics, Algorithms Combin. 26, Springer (2006), 591–609. 38. J. Širánˇ and T. W. Tucker, Characterization of graphs which admit vertex-transitive embeddings, J. Graph Theory 55 (2007), 233–248. 39. J. Širán,ˇ T. W. Tucker and M. E. Watkins, Realizing finite edge-transitive orientable maps, J. Graph Theory 37 (2001), 1–34. 40. M. Škoviera and J. Širán,ˇ Regular maps from Cayley maps, Part I: balanced Cayley maps, Discrete Math. 109 (1992), 265–276. 41. M. Suzuki, On finite groups with cyclic Sylow subgroups for all odd primes, Amer. J. Math. 77 (1955), 657–691. 42. W. T. Tutte, What is a map?, New Directions in the Theory of Graphs (ed. F. Harary), Academic Press (1973), 309–325. 43. S. E. Wilson, Cantankerous maps and rotary embeddings of Kn, J. Combin. Theory (B) 47 (1989), 262–279. 44. S. E. Wilson, Parallel products in groups and maps, J. Algebra 167 (1994), 539–546. 45. W. J. Wong, On finite groups with semi-dihedral Sylow 2-subgroups, J. Algebra 4 (1966), 52–63. 11 The genus of a group

THOMAS W. TUCKER

1. Introduction 2. Symmetric embeddings and groups acting on surfaces 3. Quotient embeddings and voltage graphs 4. Inequalities 5. Groups of low genus 6. Genera of families of groups References

This chapter surveys the genus of a ﬁnite group. Various symmetric embeddings of Cayley graphs are discussed, together with their associated genus parameters and their relationship to group actions on surfaces. Computations for low genus and certain families of groups are given. Particular attention is paid to general results relating the various genus parameters to each other.

1. Introduction

The (orientable) genus γ (A) of a finite group A is the smallest integer h such that some Cayley graph for A can be embedded in the orientable surface Sh. (Recall that the Cayley graph C(A,X) for a group A with generating set X has vertex-set A and edges between a and ax, for all a ∈ A and x ∈ X.) The term was first introduced by White [50], but similar ideas appear as far back as the late 19th century. Burnside [6] has two chapters on the ‘graphical representation of a group’ that include the determination of all groups of ‘genus’ 0 and 1 (really the strong symmetric genus, in the language of the next section). The early history is mostly in the context of finite groups of conformal automorphisms of Riemann surfaces, and this context continues to play an important role. On the other hand, Burnside

225 226 Thomas W. Tucker also viewed an embedding of a Cayley graph, or more explicitly the faces of such an embedding, as a way of understanding the relations in a group presentation, in the spirit of Dehn [14] a few years later. White’s interest stemmed from the combinatorial current graph construction of Ringel and Youngs. Thus, from its origins to its current practice, the study of the genus of a group has inevitably involved an interplay of Riemann surfaces and branched coverings, combinatorial group theory of generators and relations, and combinatorics of topological graph theory. One can also define the non-orientable genus γ (A) to be the smallest positive integer k such that some Cayley graph for A embeds in the non-orientable surface e Nk, and the Euler genus γ (A) to be the minimum of 2γ (A) and γ (A). We should observe at the outset that the genus parameter is not particularly interesting for infinite groups, since every infinite group has genus either 0 or infinity (see Levinson [23]). The underlying reason is simple enough: if an infinite Cayley graph is not embeddable in the plane, then it must contain K5 or K3,3, and this gives rise to infinitely many disjoint replicas via the natural action of the Cayley group. On the other hand, characterizing infinite planar groups with infinitely many ends is a difficult problem and is not resolved even for 2-connected Cayley graphs. In this chapter, we consider only finite groups. The outline of this chapter is as follows. Section 2 relates the genus of a group to actions of finite groups on surfaces and introduces the symmetric and strong symmetric genus and their non-orientable counterparts. In Section 3 we discuss regular voltage graphs and quotient graphs and the Riemann–Hurwitz equation. Section 4 is a survey of general theorems relating the genus of a group to the genus of a subgroup, the genus of a group to its order, and the different genera to each other. Section 5 is concerned with the determination of all groups of a given low genus for each of the different genus parameters, while Section 6 focuses on the genera of several special families of groups.

2. Symmetric embeddings and groups acting on surfaces

Throughout this chapter we consider group actions on surfaces and graphs. When we say that a group A acts on a set S we mean here that it acts faithfully – that is, the only element of A that acts as the identity on S is the identity of A. Also, all actions are by automorphisms for graphs, and by homeomorphisms for surfaces. Given the Cayley graph C(A,X), left-multiplication by a ∈ A is a graph automorphism, leaving no vertex fixed if a = 1. Indeed, if the group A acts regularly (transitively without fixed points) on a graph, then the graph is a Cayley graph for A (see Sabidussi [41]). An embedding of a Cayley graph C(A,X) in a surface S is symmetric if the natural action of A by left-multiplication on the 11 The genus of a group 227 vertices of C(A,X) can be extended to an action on the surface S. If the surface is orientable and the extended action preserves orientation, then the embedding is strongly symmetric. Alternatively, symmetric embeddings can be described in terms of the cyclic order of the edges at each vertex, the rotation system of the embedding. For an embedding in an orientable surface, a graph automorphism that respects the rotation system (that is, preserves the cyclic order at each vertex), can be extended to an orientation-preserving homeomorphism of the surface, while one that reverses the rotation system can be extended to an orientation-reversing homeomorphism of the surface (see Gross and Tucker [18]). Thus, an embedding of a Cayley graph C(A,X) in an orientable surface S is strongly symmetric if the cyclic order of generators and their inverses is the same around each vertex; this means that left- multiplication by A preserves the rotation system and hence extends to an action by A on S.Astrongly symmetric embedding of a Cayley graph is sometimes called a Cayley map and is discussed in Chapter 10. An embedding of a Cayley graph C(A,X)in an orientable surface is symmetric, but not strongly symmetric, if there is a subgroup B of index 2 in A such that all vertices labelled by elements of B have the same rotation and all other vertices in A − B have the reverse rotation. If b ∈ B, then bB = B and b(A − B) = A − B, so left-multiplication by b ∈ B preserves the rotation system; similarly, left-multiplication by a/∈ B reverses the rotation system. We conclude that the action of A by left-multiplication extends to an action of A on the surface S with the subgroup B corresponding to the elements of A whose action on the surface S preserves orientation. These embeddings have been little studied; there is no foundational reference like [40] for Cayley maps, and there is not even a name for such embeddings. For an embedding in a non-orientable surface, a rotation system must also specify the type of each edge, twisted or flat (see Chapter 1). Any graph automorphism respecting the rotation of vertices and type of edges extends to a surface homeomorphism. Thus, an embedding of a Cayley graph C(A,X) in a non-orientable surface S is symmetric if every vertex has the same rotation and all edges corresponding to the same generator x have the same type (see Tucker [49]). The symmetric genus σ (A) of the finite group A is the smallest integer h such that a Cayley graph for A has a symmetric embedding in the orientable surface Sh; the strong symmetric genus σ o(A) is defined analogously. The genus of a group studied by Burnside and others at the turn of the last century was basically the strong symmetric genus. The symmetric non-orientable genusσ (A) is the smallest positive integer k such that some Cayley graph for A embeds in the non-orientable e surface Nk. The symmetric Euler genus σ (A) is the minimum of 2σ (A) andσ (A). 228 Thomas W. Tucker

Symmetric embeddings of Cayley graphs are useful for many reasons. First, they are easier to describe than other Cayley graph embeddings since one needs only a single rotation for a strongly symmetric embedding, or a rotation and an index-2 subgroup for a symmetric embedding in an orientable surface. Second, they tend to be minimum-genus embeddings more often than one might expect. Third, they provide a concrete way of giving finite group actions on surfaces; for example, every group A, together with a cyclic order of a generating set X and its inverses, automatically defines an orientation-preserving action of A on some orientable surface. Finally, every finite group action on a surface arises this way, as the following theorem (see [45]) states. Theorem 2.1 Let the finite group A act on the surface S. Then there is a Cayley graph C(A,X) symmetrically embedded in S such that the action of A on S is an extension of left-multiplication of A on C(A,X).IfS is orientable and the given action of A on S preserves orientation, the embedding is strongly symmetric. Corollary 2.2 The symmetric genus of a group A is the smallest number h for which A acts on the orientable surface Sh. Similar statements hold for the strong symmetric genus, the symmetric non-orientable genus, and the symmetric Euler genus. This theorem is really a folk theorem and follows from the following geometrical intepretation. As shown by Kerekjarto´ and others (see [18]), any finite group action on a surface comes from some geometrical structure on the surface (spherical for S0, Euclidean for S1, and hyperbolic for Sh when h>1). This means that we can use the idea of a ‘fundamental domain’ for the action – namely a tiling of the surface (by congruent tiles or copies of a fundamental domain) that is preserved by the group action and such that the group acts regularly on the set of tiles. Placing a vertex at the centre of each tile and joining it by edges to the vertices in neighbouring tiles, we obtain a graph for which the action of A on the vertices is regular, making it a Cayley graph for A. On the other hand, a proof can be given (see [45] or [18]) using the topology of quotient surfaces instead of geometry.

3. Quotient embeddings and voltage graphs

Voltage graphs and embedded voltage graphs were introduced as a more direct and consciously topological way of viewing the dual current graph construction of Ringel, Youngs and others. In the context of this chapter, however, we can view an embedded 1-vertex or 2-vertex voltage graph simply as a diagram that determines the face structure of a symmetric embedding of a Cayley graph. Some examples show what we mean. 11 The genus of a group 229

4 6 Example 1 Let A =x,y: x = y =[x,y]=1=Z4 × Z6 and X ={x,y}. Consider the strongly symmetric embedding for the Cayley graph C(A,X) in the surface S, where every vertex has the rotation x,x−1,y,y−1. Let G be a bouquet of two circles (a 1-vertex graph with two loops). Assign a direction to each loop and label one x and the other y. The rotation x,x−1,y,y−1 defines an embedding of G in some surface T . Tracing out the faces of this embedding in the usual way, we see there are 3 faces (with boundaries x, y and xy) giving Euler characteristic ε(T ) = 1 − 2 + 3 = 2, so T is the sphere. Now let us compute ε(S). The single vertex and two edges of G give rise to 24 vertices and 48 edges in S. The x-face of the embedding becomes an x4-face, since x has order 4; there are six such faces, corresponding to the six left cosets of x A = × 3 in , a deficiency of 18 24 4 from the 24 faces we might have expected. Similarly, there are four y6 faces and four (xy)6 faces (since y and xy have order = × 5 6), both deficiencies of 21 24 6 . Thus ε(S) = ε(T ) − 3 + 5 + 5 =− , 24 24 4 6 6 12 so S is the surface of genus 7. In the language of [18], the embedding of the bouquet G with edge directions and labels is an embedded regular voltage graph with voltages in the group A. The words in x and y obtained by tracing out the faces of the embedding of G are the net voltages on the directed face boundaries of the embedded voltage graph. From a topological viewpoint, the embedding of G in T is the quotient embedding of the given strongly symmetric embedding of C(A,X) in S under the action of A on S: in the quotient surface T , each orbit of A on S is identified to a single point. The identification map is a regular branched covering with the only branch points at the centres of some faces, and the order of a branch point is the same as the order of the net voltage on the boundary of the face. From a computational viewpoint, none of this topological terminology is necessary. The main thing we need to do is to compute the Euler characteristic of S from that of T and the orders in A of the words we get when we trace out the faces of the embedding of G in T . If those face orders are r1,r2,...,rm, then we have ε(S) =|A| ε(T ) − ( − 1 ) , 1 r i i which is usually called the Riemann–Hurwitz equation. Some care needs to be taken with involutions in the generating set. If x ∈ X is an involution for a strongly symmetric embedding of C(A,X)in S, then the action of x on S looks like rotation by 180◦ about the midpoint of some edge labelled x. In this context, it is easiest to think of the edges of x as appearing in parallel 230 Thomas W. Tucker pairs, bounding a digon corresponding to x2 = 1. Thus, in the quotient embedding we should represent x by a loop that bounds a face. On the other hand, if S is orientable and the embedding of C(A,X) is symmetric, but not strongly so, then the action of an involution x may look like a reflection. The following example shows how to view such a situation. Example 2 Let A =x,y,z: x6 = y2 = z2 = 1, [x,y]=[y,z]=[x,z]= 1=Z6 × Z2 × Z2 and let X ={x,y,z}. Consider the symmetric embedding with rotation x,x−1,y,zand its reverse, with B the index-2 subgroup generated by x and y. The quotient embedding under the action of B has two vertices v and w, joined by a single z-edge with an x-loop and a y-loop at each end-point. The rotation at v is x,x−1,y,zand at w it is the reverse. There are two faces bounded by x, two by y, and one by zxyzyx, so the quotient surface T is the sphere. The two x-faces in the embedding in T correspond to x6-faces in the embedding of C(A,X) in S, the two y-faces to y2-faces (digons), and the xyzyxz-face to (xyzyxz)3-faces, since xyzyxz = x2, which has order 3 in A. In the Riemann–Hurwitz equation, we multiply by |B|=12, not by |A|=24: ε(S) = − 5 − 5 − 1 − 1 − 1 =− . 12 2 6 6 2 2 3 16

The situation for a non-strongly symmetric embedding of a Cayley graph C(A,X) in an orientable surface is the same as in this example. We take the quotient by the index-2 subgroup B to get an embedding of a 2-vertex graph G in a surface T , with the rotation at w being the reverse of that at v. Each generator x in B yields a directed loop in G at the vertices v and w, and each generator x not in B yields either an undirected edge between v and w if x is an involution, or a pair of directed edges between v and w, otherwise. If r1,r2,...,rm are the face orders (that is, the orders of the words we get by tracing out the faces of this quotient embedding), then the Riemann–Hurwitz equation is 1 ε(S) = 1 |A| ε(T ) − (1 − ) . 2 r i i

The case where T is the sphere and the quotient embedding has three faces occurs frequently. The three types of quotient embedding are shown in Fig. 1, where the orders of the face boundaries are denoted by p, q, r. These three types lead to the following kinds of groups and embeddings in surfaces of given Euler characteristic:

(p,q,r)o: A =x,y: xp = yq = (xy)r = 1,..., 1 1 1 strongly symmetric: ε(S) =|A| − + + + ; 1 p q r 11 The genus of a group 231 properly (p,q,r): A =x,y,z: x2 = y2 = z2 = (xy)p = (yz)q = (xz)r ,..., where B =xy, yz has index 2, 1 1 1 symmetric: ε(S) = 1 |A| − 1 + + + ; 2 p q r properly (p,p,r)c: A =x,y: xp = y2 =[x,y]r = 1,..., where B =x,yxy−1 has index 2 , 1 1 1 symmetric: ε(S) = 1 |A| − 1 + + + . 2 p q r For the second and third types, the subgroup B has index at most 2; if it has index 1, we omit the word ‘properly’ or use the word ‘improperly’.

r r r

p p q p p q

Fig. 1. Three quotient embeddings in the sphere

Closely related to (2,q,r)o groups are regular maps – that is, embeddings of graphs on surfaces for which the automorphism group acts transitively on the set of directed edges. Such a map can be identified with its automorphism group, so that the existence of a regular map of degree q and face size r on an orientable surface implies the existence of a (2,q,r)o group acting on the surface, preserving orientation. Regular maps of small genus have been well studied and the lists of such groups in [16], [42] and [10] are extremely useful in determining all groups of small strong symmetric genus or symmetric genus. Although we have described the quotient information for a group action in terms of an embedded voltage graph with one or two vertices, many authors use the viewpoint of conformal automorphisms, where the information is given as the signature of the quotient surface as a Riemann surface, an expression with various brackets indicating the orientability, orientable or non-orientable genus, number of boundary components (caused by reflections), orders of branch points not on the boundary, and the orders of branch points on each boundary component. The voltage graph viewpoint presents this information as a picture, but when the quotient surface is not the sphere and the action has reflections, the signature is a much more compact way of giving the quotient information. This viewpoint can be found, for example, in the papers of May and Zimmerman [27], [31]. 232 Thomas W. Tucker

4. Inequalities

In this section, we consider various inequalities for genus parameters: between a subgroup and a group, between the order of a group and its genus parameters, and between the genus parameters themselves. If a group A acts on a surface, then each of its subgroups acts on the same surface. Thus, by deﬁnition, if B is a subgroup of A, then o o e e σ(B) ≤ σ (A), σ (B) ≤ σ (A), σ(B) ≤ σ (A), and σ (B) ≤ σ (A).

On the other hand, there is no obvious reason why similar inequalities should hold for the genus, non-orientable genus, or Euler genus of groups. The following theorem of Babai [1] implies that such inequalities do indeed hold. Theorem 4.1 Let B be any subgroup of A. Then any Cayley graph for A contracts to a Cayley graph for B. Since the proof is so simple, we sketch it. Let G be the Cayley graph and let p: G → G/B be the graph covering obtained by identifying each orbit of B to a point. Then, for any spanning tree T of G/B, the lift p−1(T ) is a spanning forest for G with B acting regularly on its components. Contracting each component to a vertex, we get a graph H with A acting regularly on the vertices; this, as we have observed, makes H a Cayley graph for B. Moreover, if G embeds in a surface, so does H , so we have: Corollary 4.2 If B is a subgroup of A, then γ(B) ≤ γ (A), γ(B) ≤ γ (A) and γ e(B) ≤ γ e(A). The prototype for inequalities between the order of a group and its genus parameters, as well as the oldest theorem on the genus of a group, is due to Hurwitz [20]: Theorem 4.3 Let A be a group.

If σ (A) > 1, then |A|≤168(σ (A) − 1) and |A|≤84(σ e(A) − 2). If σ o(A) > 1, then |A|≤84(σ o(A) − 1) and |A|≤84(σ (A) − 2).

Proof The right-hand side of the Riemann–Hurwitz equation is largest, but still negative, when T is the sphere and there are three faces of orders 2, 3 and 7; that is, A is properly (2, 3, 7). Then

2 − 2σ (A) = ε ≤−|A|/84, so |A|≤168(σ (A) − 1).

The argument for σ o(A) is the same, with A being (2, 3, 7)o. The results for σ and σ e follow from the orientable cases using orientable double coverings (see [45]). 11 The genus of a group 233

Again, there is no reason to expect the analogous theorem to hold for γ,γ,γe, but Tucker has shown that it does (see [18]): Theorem 4.4 Let A be a group with γ (A) > 1. Then

e |A|≤168(γ (A) − 1), |A|≤84(γ (A) − 2), and |A|≤84(γ (A) − 2).

The proof is long. The basic idea is that if |A| is large compared with the genus of a surface containing a Cayley graph for A, then by Euler’s formula the degree of the graph must be small (generally 4 or less) and there must be many small faces. Thus A has a presentation with few generators and short relations. These short relations can then be used to construct an embedding of genus 0 or 1 for the Cayley graph. The orientability of the embedding surface does not matter, only its Euler characteristic. In fact, if γ (A) > 1 and if some Cayley graph for A can be embedded in a surface of Euler characteristic ε, then |A|≤−84ε. Hurwitz’s theorem for σ (A) can be refined when |A| > 24(σ (A) − 1) to show that A must have one of the three (p,q,r)-types with restriction on the sizes of p, q and r. The proof is just a more careful analysis of the possibilities allowed by the Riemann–Hurwitz equation (see [43], [45]). A similar refinement exists for γ (A) as well, but with a much more difficult proof (see [47]). These refinements of Hurwitz’s theorem are especially useful in determining groups of small genus or symmetric genus. Next we consider inequalities between the seven different genus parameters. Since any orientable embedding of a graph other than a tree can be made into a non-orientable embedding by ‘twisting’ a single edge, which adds a crosscap, it follows that γ (A) ≤ 2γ (A) + 1. There is no obvious inequality in the other direction, and in fact there are examples in [36] where 2γ (A) − γ (A) appears likely to be arbitrarily large. There is no similar inequality for the symmetric genus or the symmetric non- orientable genus, since twisting a single edge destroys any symmetry. On the other hand, by using orientable double coverings (a standard technique for the topology of surfaces [45]), one can show that if A acts on the non-orientable surface of Euler characteristic ε (crosscap number 2−ε), then A×Z2 acts on the orientable surface of Euler characteristic 2ε (genus 1 − ε) with the A factor preserving orientation. Thus, o σ (A) ≤ σ (A) − 1 and σ (A) ≤ σ(A× Z2) ≤ σ (A) − 1. The Hurwitz bound can be exploited to show that all of the genus parameters are related by linear inequalities when the genus is greater than 1. Moreover, these inequalities depend mostly on |A| and can be extended to obtain inequalities between the genus parameters of a group and those for its quotient groups. The 234 Thomas W. Tucker inequalities stated at the end of [49] are specific examples of our next theorem. First, let δ(γ) = δ(σ) = δ(σo) = 1 and let δ(γ)= δ(σ) = δ(γ e) = δ(σe) = 2. Theorem 4.5 Let ρ and τ be any two of the seven genus parameters (possibly ρ = τ). Then there exists a constant m, depending only on ρ and τ, such that for any group A with τ(A) > δ(τ) and any normal subgroup N of A, τ(A) − δ(τ) ρ(A/N) − δ(ρ) ≤ m . |N|

Proof We include a proof since it is not difﬁcult, it does not appear in the literature, and it illustrates the key role of Euler’s formula. We consider only the case ρ = σ o and τ = γ ; the other cases are handled in the same way. We want to ﬁnd a number m for which γ (A) − σ o(A/N) − ≤ m 1. 1 |N| Suppose that γ (A) is achieved by an embedding of the Cayley graph C(A,X) of degree d in a surface of Euler characteristic 2 − 2γ (A). By standard counting arguments, using Euler’s formula and the fact that all faces have size 3 or more, we have − γ (A) ≤|A|( − 1 d + 1 d) =|A|( − 1 d). 2 2 1 2 3 1 6 Since the image of X under the quotient map from A to A/N generates A/N, the quotient group A/N has a Cayley graph of degree d as well. That Cayley graph has a strongly symmetric embedding in some surface; the Euler characteristic of |A/N|( − 1 d) that surface is at least 1 2 (assume there are no faces at all). Thus − σ o(A/N) ≥|A/N|( − 1 d) 2 2 1 2 , and therefore

o |A|(d − 2) σ (A/N) − 1 ≤ . 4|N| First assume that d>6. Rewriting the inequality for γ (A), we have 2γ (A) − 2 γ (A) − 1 |A|≤ = 12 . 1 d − d − 6 6 1 Thus, γ (A) − σ o(A/N) − ≤ c 1, 1 3 |N| where c = (d − 2)/(d − 6). Since d>6, we have c ≤ 5, so

o σ (A/N) − 1 ≤ m(γ (A) − 1)/|N| with m = 15.

Assume now that d ≤ 6. Then σ o(A/N) − 1 ≤|A|/|N|. By Theorem 4.4, |A|≤168(γ (A) − 1),soσ o(A/N) − 1 ≤ m(γ (A) − 1)/|N|, with m = 168. Choosing the larger m from the two cases, we have the desired inequality. 11 The genus of a group 235

The situation when ρ = σ involves a little care. It is not true that every Cayley graph has a non-orientable symmetric embedding; if the graph is bipartite, then twisting all the edges for all the generators still does not give a non- orientable embedding. One can always, however, add to a generating set X one extra generator, which is the product of two generators already in X, to destroy bipartiteness and allow a non-orientable symmetric embedding of the resulting Cayley graph. This increases the degree for the ρ inequalities from d to d + 2, but it does not affect the existence of m. (This observation was not made in [49], where the existence of m in the case when ρ = σ was left as an open question). For given genus parameters ρ and τ, we would like upper bounds on the least value m(ρ, τ), for the constant m guaranteed by the theorem. For example, the preceding proof shows that m(σ, γ ) ≤ 168. For the case d<6, the reﬁned Hurwitz theorems can be used to get better estimates for m(ρ, τ), since a ‘good’ (p,q,r) presentation for A guarantees one as well for A/N.The case d ≥ 6 can be improved by counting how many triangles can be incident to a vertex (there are no relations of the form x2y = 1orxyz = 1, since it always pays to eliminate redundant generators). It appears that lengthy analysis of this sort can get m(σ, γ ) < 4. In [49], m(σ, γ ) < 6 is claimed, but without proof. The case of ρ = σ o and τ = σ appears to require the largest value for m. Of particular interest is the case ρ = τ, since then we are comparing the values of a genus parameter for a group and a quotient of that group. For a symmetric genus, this is not difﬁcult. For example, if A acts preserving orientation on the surface S of Euler characteristic ε, then A/N acts preserving orientation on the quotient surface S/N, whose Euler characteristic, by the Riemann–Hurwitz equation, is greater than ε/|N| (for details, see [45]). Thus, σ o(A/N) − 1 ≤ (σ o(A) − 1)/|N|, so here m ≤ 1. Orientation-reversing actions and non-orientable surfaces can introduce a factor of 2, but in general m(ρ, ρ) ≤ 2 for any of the symmetric genera, so ρ(A/N) ≤ ρ(A), since |N|≥2. On the other hand, it is still an open question whether γ(A/N)≤ γ (A).Ifm(γ, γ ) can be reduced to 4, then the only issue would be normal subgroups of order 2 or 3. Another consequence of the theorem is that all the different genus parameters are asympotic to each other, once one discards groups of genus 0 or 1. Thus, all the genus parameters behave qualitatively the same for groups of genus 2 or more as the size of the groups gets large.

5. Groups of low genus

The ﬁrst important result on the genus of a group is Maschke’s characterization [26] of the groups of genus 0. We denote the symmetric group on n symbols by Sn, the alternating group by An, the dihedral group of order 2n by Dn and the 236 Thomas W. Tucker

cyclic group of order n by Zn. The Cartesian (direct) product of groups A and B is denoted by A × B.

Theorem 5.1 If γ (A) = 0, then A is a subgroup of the automorphism group of a Platonic solid or a prism – that is, Z2 × Dn, Z2 × S4 or Z2 × A5. In particular, γ (A) = 0 if and only if σ (A) = 0. Also, σ o(A) = 0 if and only if A is a subgroup of Dn, S4 or A5.

That γ (A) = 0 if and only if σ (A) = 0 follows from Whitney’s uniqueness theorem for 3-connected planar graphs [51]: given any embedding of C(A,X) in the sphere, the natural action of A extends to an action on the sphere. The rest of the theorem follows directly from the Riemann–Hurwitz equation which gives a short list of possible (p,q,r)types for A, as well as the order of A. It is not hard then to identify these groups; for example, x = (12), y = (34), z = (23) gives a proper (2, 3, 3) generating set for S4. For details, see Gross and Tucker [18]. The problem of classifying groups of genus 1 was first undertaken by Baker [3]. Proulx completed the classification in her thesis [38] by giving partial presentations for all such groups. These presentations fell into more than twenty different infinite classes of groups, together with four ‘sporadic’ groups. She observed that the infinite classes all seem to correspond to the presentations for the seventeen Euclidean space groups given in Coxeter and Moser [13] – that is, each class is a finite quotient of one of the seventeen space groups. Tucker [46] showed subsequently that one of the four sporadic groups was also a space group quotient but the other three were not. He also showed that σ (A) = 1 if and only if A is a space group quotient (see [45] or [18]), although this is probably a folk theorem for topologists. The proof is not structurally graph theoretic. First, the partial presentations for the groups of symmetric genus 1 can be tabulated using the Riemann–Hurwitz formula; one can easily compute that the face orders are (2, 2, 2, 2), (2, 4, 4), (3, 3, 3) or (2, 3, 6), and then one simply considers all possible 1-vertex or 2- vertex quotient graphs with these face orders. The resulting partial presentations coincide with the standard space group presentations derived from the geometry of Euclidean isometries. Knowing what presentations to expect, one then looks at a Cayley graph embedded in the torus. Assume that the generating set for the Cayley graph is irredundant – that is, no proper subset generates the group. The formula for the Euler characteristic then guarantees a degree of 3 or 4 for the embedded Cayley graph and the existence of many small faces. Thus we have a small generating set with many relations. A lengthy case-by-case analysis leads to a variety of special group presentations that are then shown to coincide with those for the groups of symmetric genus 1, with three exceptions. Showing that the three groups with exceptional presentations are truly exceptional involves some 11 The genus of a group 237

technical exploitation of the algebraic structure of the Euclidean space groups. For details and the list of presentations, see [18]. We summarize:

Theorem 5.2 With three exceptions, γ (A) = 1 if and only if σ (A) = 1.In particular, if γ (A) = 1, then, with three exceptions, A has a partial presentation making it a finite quotient of one of the seventeen Euclidean space groups (for each space group, there are infinitely many finite quotients). The three exceptional groups have the presentations:

(a)A=x,y: x3 = y3 = 1,xyx = yxy; (b)A=x,y: x3 = y2 = 1,xyxyxy = yxyxyx; (c)A=x,y: x3 = y2 = 1, (xyxyx−1)2 = 1.

Group (a) has order 24, while groups (b) and (c) have order 48. Group (c) is isomorphic to the group GL(2, 3) of invertible 2 × 2 matrices with entries in Z3 and group (a) is the subgroup SL(2, 3) of matrices with determinant 1. The situation for groups of genus 2 is completely different. By the Hurwitz bounds, the order of any group of genus g>1 is bounded by 168(g − 1), so there can be only a ﬁnite number of groups of genus 2. In fact, as shown by Tucker [48], there is only one:

Theorem 5.3 The only group of genus 2 has order 96 and the presentation

A =x,y,z: x2 = y2 = z2 = (xy)2 = (yz)3 = (xz)8 = 1, [(xz)4,y]=1.

It is not hard to see why the presentation for A gives a group of order 96: the involution (xz)4 is central and its quotient is the familiar proper (2, 3, 4) presentation for Z2×S4 that appears in Maschke’s theorem for planar groups.Also, the group is properly (2, 3, 8), so it has a symmetric embedding in the orientable ( 96 )(− + 1 + 1 + 1 ) =− γ (A) ≤ surface of Euler characteristic 2 1 2 3 8 2, so 2. It can be shown that toroidal group (c) is isomorphic to the subgroup generated by xy and yz,soσ (A) = 1 and hence γ (A) = 1, since |A|=96; thus γ (A) = 2. Showing that A is the only group of genus 2 involves a lengthy analysis of all groups of order less than 96, with partial presentations restricted by the reﬁned Hurwitz theorem for γ . Inspired and helped by Tomaž Pisanski, Dewitt Godfrey created a sculpture of the symmetric embedding of C(A,{x,y}) in S2, which was installed in June 2007 at the Technical Museum in Bistra, Slovenia (for a photograph, see [32]). The group A of genus 2 appears in a number of group-theoretic contexts (see [12]). Since the subgroup generated by xy and yz, which is isomorphic to group (c), is the orientation-preserving subgroup of the action of A on the surface of genus 2, it has symmetric genus and strong symmetric genus 2, as does the exceptional group (a), since it is a subgroup of group (c). The subgroup of A generated by yz and x is 238 Thomas W. Tucker isomorphic to the exceptional group (b), but does not preserve orientation; it has symmetric genus 2 but strong symmetric genus 4. In fact, May and Zimmerman ([28]) have shown that this single group A contains all the groups of symmetric genus 2:

Theorem 5.4 The group of genus 2 and the three exceptional groups of genus 1 are the only groups of symmetric genus 2.

Table 1 summarizes what is presently known about genus 2, 3 and 4 for γ,σo and σ . Where the total number of groups is known, we give the number and a reference; otherwise, we list groups known to have the given genus. Notice that σ (A) = 3 implies γ (A) = 3, since γ (A) ≤ σ (A) and, for γ (A) < 3, we have σ (A) = γ (A) for all A except toroidal groups (a) – (c), which have σ (A) = 2. We let Q denote the quaternion group of order 8.

Table 1. Groups of low genus

o genus γσσ

2 one [48] six [30] four [28] 3 σ = 3, Z2 × Q [50], more? ten [31] three [28] 4 S5 [39], more? ten [31] —

For small non-orientable genera, the most interesting question is whether some groups have their Euler genus embedding in a non-orientable surface. For the projective plane, the answer is yes. In [18], it was claimed that Z3 × Z3 is the only group of Euler genus 1. In fact, there is one other (discovered by an undergraduate student [19]): the group A =x,y: x3 = y2 =[x,yxy]=1. Notice that the subgroup B =x,yxy is Abelian, and hence isomorphic to Z3 × Z3; since it is also normal of index 2, |A|=18. If the six x3 triangles in the Cayley graph C(A,{x,y}) are contracted to a point, the resulting graph is easily seen to be K3,3. Since K3,3, as a hexagon with edges joining antipodal vertices, embeds in the projective plane, so does C(A,{x,y}). Since Z3 × Z3 is not in the list of Theorem 5.1, neither is A, so both groups have Euler genus 1. The genus is 1 for both groups (for A, see Theorem 6.3.3(n) of [18]). Table 2 summarizes what is known about γ,σ,γe and σ e for the values 1, 2, 3, 4 and 5. If a box has a question mark together with a reference, it means that the techniques in the reference appear likely, under careful inspection, to apply to the given case. The results for σ e follow directly from orientable double coverings, as does the case σ = 1. 11 The genus of a group 239

Table 2. Groups of low non-orientable and Euler genus

e e genus γ σγσ e o 1 γ = 1, γ = 0 σ = 0, but not A4 two [18, 19] none 2 none? [49] Z2 × Zn, Dn × Z2 [49] γ = 1 [18] σ = 1 3 γ = 1? [48] none [30] none? [48] none [30] 4 none? [48] none? γ = 2 [48] σ = 2 [30] 5 S5, more? S5, more? S5, more? S5 [30]

There is one other result about embeddings of low genus that, even if peripheral, deserves special mention. Although we know that there are only finitely many groups of a given genus h>1, and by [47] only finitely many irredundant Cayley graphs of that genus, there could be infinitely many groups of genus at most 1 with (redundant) Cayley graphs of genus h. In fact, that does happen for h = 2: 2n 2 n if A = Z2n × Z2 =x,y: x = y = x,y = 1, then C(A,{x,y,x }) can be embedded in S2 but not in the torus (for details, see [47]). On the other hand, this class of Cayley graphs is, in effect, the only example, as proved independently by Babai [2] and Thomassen [44]: Theorem 5.5 There are only finitely many vertex-transitive graphs of a given genus h>2. In particular, there are only finitely many Cayley graphs of a given genus h>2. Both Babai and Thomassen provide bounds on the number of vertices in a vertex-transitive graph of genus h>2, but the bound seems much too large. Indeed, we would expect the bound not to exceed the Hurwitz bound of 168(h−1), which is the case (for example) when the automorphism group of the graph extends to the embedding surface.

6. Genera of families of groups

In this section we survey what is known about the different genus parameters for various families of groups. We focus first on groups of low order. Proulx [38] found γ for all groups in Coxeter and Moser’s list ([13]) of groups of order less than 32, except Z3 × Z3 × Z3 and another group of order 27; it was discovered only later that the second group had been omitted by Proulx. The case Z3 × Z3 × Z3 is discussed below; the other group of order 27 is handled in [5]. Thus, the genus of every group of order less than 32 is known. The obvious infinite family to study first is that of Abelian groups, but the situation is very complicated, since minimum embeddings for γ are not likely to be symmetric. Here is why. If A is Abelian, then any Cayley graph for A will 240 Thomas W. Tucker have lots of 4-cycles, corresponding to the different generators commuting with each other. Thus one might expect a minimum embedding to have all of its faces quadrilaterals. But if x and y are generators of A and we want xyx−1y−1 to bound a face, then the four rotations at the vertices around that face must take x to y, y−1 to x, x−1 to y−1 and y to x−1. If we want the same rotation or its reverse at every vertex, then there is no room in the rotation for other generators, and thus the embedding cannot be symmetric. Nevertheless, it is not hard to construct a quadrilateral embedding for any Cartesian product of even cycles (by the tubing or patchwork constructions of [50] and [18]); this gives minimum embeddings when A is the Cartesian product of even-order cyclic groups. In general, we would like to assert that an Abelian group has a quadrilateral embedding whenever Euler’s equation allows it. What we can say is this. Write A × ×···× m > the Abelian group in canonical form Zm1 Zm2 Zmr , where 1 1 and mi divides mi+1 for all i. Then Jungerman and White [22], in what is still the greatest tour de force of the original Ringel–Youngs current graph construction, proved the following result.

m r> γ (A) = + 1 |A|(r − 1 s − ) Theorem 6.1 If 1 is even and 2, then 1 4 2 2 , where s is the number of Z2-factors in the canonical form for A.Ifm1 > 3 is odd, if r> |A|(r − ) γ (A) = + 1 |A|(r − ) 3, and if 4 divides 2 , then 1 4 2 .

The presence of Z3-factors complicates the issue enormously, since this allows the possibility of triangular faces. We have already seen that Z3 × Z3 is one of the two groups of Euler genus 1. The case of Z3 × Z3 × Z3 is even more notorious. Although the symmetric genus is easily computed to be 10, Mohar, Pisanski, Škoviera and White [33] gave a surprisingly non-symmetrical embedding of genus 7. Brin and Squier [4] followed with an ingenious exhaustive proof that this is best possible:

Theorem 6.2 γ(Z3 × Z3 × Z3) = 7.

Nothing else is known about other Abelian groups with Z3-factors in their canonical forms. For more examples of bizarre embeddings for Z3 × Z3 and Z3 × Z3 × Z3, see [18]. Since commutator-based quadrilateral embeddings for Abelian groups are not symmetric for genus h>1, we expect σ o and σ to behave differently from γ ; they have been determined for all Abelian groups, respectively, by Maclachlin [25] and by May and Zimmerman [27]. Even non-Abelian groups can have many commuting generators, and the current-graph constructions of [22] do not require that all the generators commute with each other. This observation was exploited by Pisanski and Tucker [35] to compute the genus of γ(G× A) for all groups G and most Abelian groups A with 11 The genus of a group 241 sufficiently large rank. We call an Abelian group JW if it satisfies the hypotheses of Theorem 6.1. Theorem 6.3 Let G be a finite group and let H = G/[G, G] be its Abelianization. Let A be an Abelian group for which the rank r of H ×A is at least twice the rank of G H ×A γ(G×A) = + 1 |G×A|(r − 1 s) + 1 |G×A|(r − ) .If is JW, then 1 4 2 or 1 4 2 , depending on whether m1 is even or odd, respectively. Hamiltonian groups (groups for which all subgroups are normal) have the form × s × B B Q Z2 , where is an Abelian group of odd order, so Theorem 6.3 can be applied to determine γ for most such groups (see [34]). The refined Hurwitz theorems provide useful lower bounds for (2,q,r)o and (2,q,r)groups, and a surprisingly large number of groups have such generating sets. For example, a consequence of the classification of simple groups is the fact that all simple groups are (2,q,r)o. One case of particular interest is when A is properly (2, 3, 7), since then we know that γ (A) = σ (A) = 1 +|A|/168 and γ e(A) = σ e(A) = 2 +|A|/84. Similarly, if A is (2, 3, 7)o, we know that σ o(A) = 1 +|A|/84; for this reason, (2, 3, 7)o groups are called Hurwitz groups [9]. One of the first major results on Hurwitz groups is by Conder [7], [8] for the alternating and symmetric groups.

Theorem 6.4 For n>167, the symmetric group Sn is properly (2, 3, 7) and the alternating group An is improperly (2, 3, 7). e e e e For n>167, this determines γ,γ ,σ and σ for Sn and γ,γ ,σ and σ for o An. It also determines σ for An, since an improper (2, 3, 7) generating set x,y,z o gives the (2, 3, 7) generating set xy, yz. Since An is simple, it has no index-2 o subgroups, so σ = σ . This leave only γ(An): its determination appears to be very difficult. e e Theorem 6.4 also determines γ(Sn), since γ ≤ γ + 1 and γ(Sn) = γ (Sn) = 2 +|Sn|/84 implies that Sn has an improper (2, 3, 7) generating set x,y,z; this is impossible since the relations (yz)3 = 1 and (xz)7 = 1 imply that yz and xz are even permutations, so yz,xz=xy, yz = Sn. We observe that Z2 × An behaves just like Sn, since an improper (2, 3, 7) generating set x,y,z for An gives a proper (2, 3, 7) generating set (1, x), (1, y), (1,z)for Z2 × An. Conder [8] has also determined the symmetric genus of An and Sn for all n<168. Anumber of authors have determined the strong symmetric genus, and hence the symmetric genus, for various families of simple groups: sporadic groups, twelve of which are Hurwitz (see, for example, [11], [21] and [52]); PSL(2,q)(see [17]); other simple groups of Lie type having large enough dimension are Hurwitz (see, for example, [24]). Finally, we note that groups with σ = 1 can have arbitrarily large values for σ o o orσ. One such family is Dm ×Zn. May and Zimmerman [31] computed σ for this 242 Thomas W. Tucker family, with the important consequence that, for every h>1, there is a group A in this family with σ o(A) = h – that is, there are no gaps in the spectrum of values for σ o. Similarly, Gordejeula and Martinez [15] computed σ for this family. This fills in only three-quarters of the possible values for σ . It is not known whether γ or σ have gaps. For example, using a patchwork construction, it is easy to show that γ(Z2 × Z2m × Z2n) = 1 + mn, where m divides n and m>1. Thus the only possible gaps h for γ occur when h − 1 is square-free. Similarly, unpublished work of Conder and Tucker fills in about 90 per cent of the possible values for σ.

References

1. L. Babai, Some applications of graph contractions, J. Graph Theory 1 (1977), 125–130. 2. L. Babai, Vertex-transitive graphs and vertex-transitive maps, J. Graph Theory 15 (1991), 587–627. 3. R. P. Baker, Cayley diagrams on the anchor ring, Amer. J. Math 53 (1931), 645–669. 4. M. G. Brin and C. C. Squier, On the genus of Z3 × Z3 × Z3, European J. Combin. 9 (1988), 431–443. 5. M. G. Brin, D. E. Rauschenberg and C. C. Squier, On the genus of the semi-direct product of Z9 by Z3, J. Graph Theory 13 (1989), 49–61. 6. W. Burnside, Theory of Groups of Finite Order, Cambridge University Press, 1911. 7. M. D. E. Conder, Generators of the alternating and symmetric groups, J. London Math. Soc. (2) 22 (1980), 75–86. 8. M. D. E. Conder, The symmetric genus of alternating and symmetric groups, J. Combin. Theory (B) 39 (1985), 179–186. 9. M. D. E. Conder, Hurwitz groups: a brief survey, Bull. Amer. Math. Soc. 23 (1990), 359–370. 10. M. D. E. Conder and P. Dobcsányi, Determination of all regular maps of small genus, J. Combin. Theory (B) 81 (2001), 224–242. 11. M. D. E. Conder, R. A. Wilson and A. J. Woldar, The symmetric genus of sporadic groups, Proc. Amer. Math. Soc. 116 (1992), 653–663. 12. H. S. M. Coxeter, The group of genus two, Rend. Sem. Mat. Brescia 7 (1982), 219–248. 13. H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups (4th edn.), Springer, 1980. 14. M. Dehn, Papers on Group Theory and Topology (transl. J. Stillwell), Springer, 1987. 15. J. J. Etayo Gordejeula and E. Martinez, The Symmetric Crosscap Number of the Groups Cm × Dn, preprint, 2007. 16. D. Garbe, Über die regulären Zerlegungen geschlossener orienterbarer Flächen, J. Reine Angew. Math. 237 (1969), 39–55. 17. H. H. Glover and D. K. Sjerve, The genus of PSL2(q), J. Reine Angew. Math. 380 (1987), 59–86. 18. J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, 1987, and Dover, 2001. 19. J. Hannon, Senior Thesis, Colgate University, 1990. 20. A. Hurwitz, Über algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1893), 403–442. 11 The genus of a group 243

21. G. A. Jones and S. A. Silver, Suzuki groups and surfaces, J. London Math. Soc. (2) 48 (1993), 117–125. 22. M. Jungerman and A. T. White, On the genus of finite Abelian groups, European J. Combin. 1 (1980), 243–251. 23. H. Levinson, On the genera of graphs of group presentations, Ann. New York Acad. Sci. 175 (1970), 277–284. 24. A. Luchini, M. C. Tamburini and J. S. Wilson, Hurwitz groups of large rank, J. London Math. Soc. (2) 61 (2000), 81–92. 25. C. Maclachlin, Abelian groups of automorphisms of compact Riemann surfaces, Proc. London Math. Soc. 15 (1965), 699–712. 26. H. Maschke, The representation of finite groups, Amer. J. Math. 18 (1896), 156–194. 27. C. L. May and J. Zimmerman, The symmetric genus of finite Abelian groups, Illinois J. Math. 37 (1993), 400–423. 28. C. L. May and J. Zimmerman, The groups of symmetric genus three, Houston J. Math. 23 (1997), 573–590. 29. C. L. May and J. Zimmerman, Groups of small strong symmetric genus, J. Group Theory 3 (2000), 233–245. 30. C. L. May and J. Zimmerman, The group of Euler characteristic −3, Houston J. Math. 27 (2001), 737–752. 31. C. L. May and J. Zimmerman, There is a group of every strong symmetric genus, Bull. London Math. Soc. 35 (2003), 433–439. 32. The Möbius–Kantor graph, http://en.wikipedia.org/wiki/Möbius- Kantor_graph. 33. B. Mohar, T. Pisanski, M. Škoviera and A.T. White, The cartesian product of three triangles can be embedded into a surface of genus 7, Discrete Math. 56 (1985), 87–89. 34. T. Pisanski and T. W. Tucker, The genus of low rank Hamiltonian groups, Discrete Math. 78 (1989), 157–167. 35. T. Pisanski and T. W. Tucker, The genus of a product of a group with an Abelian group, European J. Combin. 10 (1989), 469–475. 36. T. Pisanski, T. W. Tucker and D. Witte, The non-orientable genus of some metacyclic groups, Combinatorica 12 (1992), 1–11. 37. T. Pisanski andA. T. White, Non-orientable embeddings of groups, European J. Combin. 9 (1988), 445–461. 38. V. K. Proulx, Ph.D. thesis, Columbia University, 1977. 39. V. K. Proulx, On the genus of symmetric groups, Trans. Amer. Math. Soc. 266 (1981), 531–538. 40. B. Richter, J. Sirá˘ n,˘ R. Jajcay, T. Tucker and M. Watkins, Cayley maps, J. Combin. Theory (B) 95 (2005), 189–245. 41. G Sabidussi, On a class of fixed-point free graphs, Proc. Amer. Math. Soc. 9 (1958), 800–804. 42. F. A. Sherk, The regular maps on a surface of genus three, Canad. J. Math. 11 (1959), 452–480. 43. D. Singerman, Symmetries of Riemann surfaces with large automorphism group, Math. Ann. 210 (1974), 17–32. 44. C. Thomassen, Tilings of the torus and the Klein bottle and vertex-transitive graphs on fixed surfaces, Trans. Amer. Math. Soc. 323 (1991), 89–105. 45. T. W. Tucker, Finite groups acting on surfaces and the genus of a group, J. Combin. Theory (B) 34 (1983), 82–98. 244 Thomas W. Tucker

46. T. W. Tucker, On Proulx’s four exceptional toroidal groups, J. Graph Theory 8 (1984), 29–33. 47. T. W. Tucker, A reﬁned Hurwitz theorem for imbeddings of irredundant Cayley graphs, J. Combin. Theory (B) 36 (1984), 244–268. 48. T. W. Tucker, There is one group of genus two, J. Combin. Theory (B) 36 (1984), 269–275. 49. T. W. Tucker, Symmetric embeddings of Cayley graphs in non-orientable surfaces, Graph Theory, Combinatorics, and Applications, Wiley-Interscience (1991), 1105– 1120. 50. A. T. White, Graphs, Groups and Surfaces (revised edn.), North-Holland, 1984. 51. H. Whitney, Non-separable and planar graphs, Trans. Amer. Math. Soc. 34 (1932), 339–362. 52. R. A. Wilson, The Monster is a Hurwitz group, J. Group Theory 4 (2001), 367–374. 12 Embeddings and geometries

ARTHUR T. WHITE

1. Introduction 2. Surface models 3. Projective geometries 4. Afﬁne geometries 5. 3-conﬁgurations 6. Partial geometries 7. Regular embeddings for PG(2, n) 8. Problems References

Finite geometries can be modelled by embedding associated graphs on surfaces, with vertices representing points and certain region boundaries representing lines. It is natural to seek embeddings of maximum efficiency (with largest possible Euler characteristic) and with interesting symmetries. We focus on projective and affine geometries, 3-configurations and partial geometries.

1. Introduction

Our objective in this chapter is to model finite geometries geometrically, with points and lines depicted as 0-dimensional and 1-dimensional subspaces of Euclidean 3-space or 4-space. We seek to do so efficiently, perhaps subject to displaying attractive symmetries, and we use surfaces for this purpose. For more information regarding the embedding of finite geometries on surfaces than can be provided here, see [19, Chap. 15]. We motivate our approach by the following example. One of the most famous and easily recognizable figures in combinatorics is the model for the Fano plane shown in Fig. 1.

245 246 Arthur T. White

3 1 6

5 0 4

Fig. 1. The Fano plane

The usual axiom system for this ﬁnite geometry is complete, so that all models for it are isomorphic. Nevertheless, this representation has several defects:

• the line {0, 1, 3} is differently shaped, yet this line is indistinguishable from the others via the axioms; the collineation group is line-transitive, as the action of the cyclic group Z7 readily shows; • each of the other lines has two ‘end’ points and one ‘middle’ point, and yet there is no concept of ‘betweenness’ in this geometry; • similarly, the point 6 seems to occupy a central position, yet this too is misleading since Z7 also establishes point-transitivity of the geometry; • we cannot discern that each point lies on exactly three lines by looking at small neighbourhoods of the points 0, 1, 3 and 6; • there are three ‘crossings’ of lines that have no meaning in the geometry.

An alternative depiction of the model for the Fano plane appears below. This regards the Fano plane as a block design, speciﬁcally as a (7, 7, 3, 3, 1)-BIBD or a Steiner triple system with 7 points. This remedies all of the above defects except the second, and even this is remedied if we represent each line abc as the set {a, b, c}.

0123456 1234560 3456012

In order to remedy the defects in a geometrical manner, we turn to models constructed as embeddings of graphs on surfaces. For example, Fig. 2 models the Fano plane by embedding the graph K7, regarded as a Cayley graph for the group 12 Embeddings and geometries 247

7 : 1

2 3 1 3

Fig. 2. The Fano plane as a graph embedding

Z7, in the torus. This embedding is constructed as the lift of the voltage graph embedding also depicted, with one vertex and three loops on the torus, using Z7 and generators 1, 2 and 3. The vertices of K7 depict the points of the geometry; the unshaded regions depict the lines (via the 3-cycles bounding those regions), and the shaded regions are the hyper-regions that remain after the graph and the line-regions are removed. Voltage graph constructions are used heavily in this chapter. Dual to current graph constructions (as in Ringel [13]), they were introduced in Gross [9] and Gross and Tucker [10]; see also [19, Chap. 10] for an expository treatment. 248 Arthur T. White

Similar models are possible for every finite geometry. We focus on two generalizations of the model just given: • the finite projective planes PG(2, q), where q is a prime power (the Fano plane is the case q = 2), and the related affine planes AG(2, q); • 3-configurations: finite geometries corresponding to designs (v, b, r, k) with k = 3, satisfying the axiom that two points lie on at most one line. Among these are the familiar geometries of Fano, Pappus and Desargues, as well as certain partial geometries, including three generalized quadrangles. We also briefly consider regular embeddings of PG(2, q). For further background and more detailed constructions, see White [18], [20], [21] and Fink and White [7].

2. Surface models

The surfaces employed here are generally closed 2-manifolds, either orientable or non-orientable. Recall that the former are classified as spheres Sh with h (≥ 0) handles attached; these exist in R3. The latter are classified as spheres Nk with k (> 0) crosscaps attached; these exist in R4, but not in R3. We also construct models in pseudosurfaces in which finitely many identifications, each of finitely many points, are made on a surface, and in generalized pseudosurfaces in which finitely many identifications, each of finitely many points, are made on a topological space consisting of finitely many components, each a pseudosurface, with a connected topological space resulting. For example, the generalized pseudosurface T = ((S4 ∪ (5S0;12(3)); 15(2))) results from first making twelve identifications of three points each to connect five mutually disjoint spheres, and then connecting the resulting space to S4 by fifteen identifications of two points each. (In Section 6 we use T to model a geometry extending the Cremona–Richmond geometry.) We refer to the models we seek as topological models. There are two graphs associated in a natural manner with a finite geometry on a point-set P and line-set L (see Coxeter [4]). We regard each l ∈ L as a subset of P , so that L is a subset of the power set of P . The Levi graph G(Ge)ofthe geometry Ge = (P , L) is the bipartite graph with partite sets P and L, and whose edges are all pairs of the form pl, where p ∈ P , l ∈ L and p ∈ l. The Menger graph M(Ge) has vertex-set P ; its edges are all pairs of the form p1p2, where p1 and p2 are collinear. Many of the graphs we embed, such as K7 in Fig. 2, are Menger graphs for the geometries they model. They are often (but not always) Cayley graphs. Each graph embedding we construct can be readily modified to give an embedding of the corresponding Levi graph on the same topological space, as illustrated for the Fano plane in Fig. 3. 12 Embeddings and geometries 249

450 2 602

124 4

6 346

561 1

3 013

235 5

Fig. 3. The Levi graph for the Fano plane

The modiﬁcation is a three-stage process:

• place a line-vertex in the interior of each line-region; • add an edge joining each line-vertex to each point-vertex in its region; • delete all the original edges.

Note that we obtain a hexagonal embedding for the Levi graph of the Fano plane. This is no accident.

Theorem 2.1 If Ge is a geometry in which two points belong to at most one line, then the Levi graph G(Ge) has girth at least 6.

Proof Since the Levi graph is bipartite, we need show only that it contains no 4- cycles. But a 4-cycle would produce two points belonging to at least two common lines, so there can be none.

Thus any hexagonal embedding of G(Ge) maximizes the Euler characteristic for all 2-cell embeddings of the geometry Ge. Such an embedding can be readily modified, if we reverse the three-step process above, to model Ge with all hyper- regions triangular. This achieves our objective of maximum efficiency for our model – that is, the maximum Euler characteristic of the ambient space. In particular, the genus γ (Ge) of the finite geometry Ge is defined to be the genus γ (G(Ge)) of the Levi graph of Ge. We have thus shown that the Fano plane has genus 1. 250 Arthur T. White

3. Projective geometries

Walsh [17] was the first to note that the Fano plane has genus 1. If PG(2, n) denotes the classical finite projective plane of order n (defined below), then the Fano plane is PG(2, 2). Walsh posed the problem of finding γ(PG(2, n)) in general. In this section we outline the solution to this problem, as presented in White [16]. We begin with some background material on these geometries. The axioms for a projective plane = (P, L) are as follows:

(1) Any two distinct points of P lie on a unique common line. (2) Any two distinct lines of L contain a unique common point. (3) There exist four points, no three of which are collinear.

It follows from these axioms that all lines have the same cardinality. We say that is ﬁnite if P is ﬁnite. Then there is a positive integer n so that, for each l ∈ L, |l| = n + 1; n is the order of = (n). It is then straightforward (see, for example, White [19, Chap. 15]) to establish the following result.

Theorem 3.1 If = (P, L) is a ﬁnite projective plane of order n, then:

• |P | = n2 + n + 1; • |L| = n2 + n + 1; • each point lies on exactly n + 1 lines; • each line contains exactly n + 1 points.

Corollary Each projective plane (n) is a symmetric balanced incomplete block design, with v = b = n2 + n + 1, r = k = n + 1 and λ = 1.

Here are some additional facts about ﬁnite projective planes (see [19, Chap. 15]); a plane is Desarguesian if it satisﬁes Desargues’ theorem, as in Euclidean geometry.

• No projective plane of order n is known to exist for n other than a prime power; it is known that projective planes of orders 6 and 10 do not exist. • If n is a prime power, there is a classical Desarguesian projective plane of order n, denoted by PG(2, n). • For n = 2, 3, 4, 5, 7 and 8, the only projective plane of order n is PG(2, n). • There are four non-isomorphic planes of order 9, including PG(2, 9). • For p a prime, m ≥ 2 and pm ≥ 9, there always exists a non-Desarguesian plane of order pm.

We concentrate on ﬁnding topological models for PG(2, n), where n is a prime power. The case n ≡ 2 (mod 3) is especially nice. We have discussed the Fano plane (n = 2) already, so next we consider PG(2, 5). 12 Embeddings and geometries 251

Commencing with GF(5) ={0, 1, 2, 3, 4} and the irreducible polynomial f(x)= x3 −x −2 over GF(5), we set x3 = x +2 and calculate increasing powers of x: 1,x, x2, x3 = x + 2, x4 = x2 + 2x, ..., x8 = x + 3, ..., x12 = 2x + 2, ..., x18 = 3x + 2, ..., x31 = 2. 62 93 124 ∼ 3 Thus x = 4, x = 3 and x = 1. It follows that x = Z124 = GF(5 ) −{0}, the multiplicative group of the degree-3 extension, and x is primitive. We now take {xi: 0 ≤ i ≤ 30} as coset representatives for (GF(53) −{0})/ (GF(5) −{0}) and Z31 for the points of (5) = PG(2, 5). (Multiplying powers of x corresponds to adding exponents.) By a theorem of Singer [14], each line in the geometry (and all its translates by the action of Z31) gives a perfect difference set: each element of Z31 −{0} occurs exactly once as a difference of distinct elements in the set. We take the unique line, the standard line l0, consisting of all those points 2 with 0 as their coefﬁcient of x . In the above calculation, l0 ={0, 1, 3, 8, 12, 18}. This generates a (31, 31, 6, 6, 1)-BIBD, which is PG(2, 5). This construction generalizes to all prime powers n. Next, we employ a crucial result that appears in Anderson [1, Theorem 2.5.2]. m 2 Theorem 3.2 Let p be prime, n = p ,v= n + n + 1, and k = n + 1. Let l0 be the standard line for PG(2, n), and let s be the sum of the elements of l0 in Zv. Let j =−s/k, with lj = l0 + j. Then multiplication by p ﬁxes the line lj .

For n = 5, j =−(0 + 1 + 3 + 8 + 12 + 18)/6 =−7 = 24, in Z31. We conﬁrm that multiplication by 5 ﬁxes l24 ={24, 25, 27, 1, 5, 11}. Rewriting l24 by orbits under this action, we obtain

l24 ={1, 5, 25}∪{11, 24, 27}={1, 11, 5, 24, 25, 27}, by interlacing the two sets. Using successive differences, we form ={10, −6, −12, 1, 2, 5} as a generating set for Z31. We now embed the Cayley graph G(Z31), in order to model PG(2, 5). Observe that 10−12+2 = 0 and −6+1+5 = 0, forcing 10−6−12+1+2+5 = 0. To see why that might work in general (and it does), note, for instance, that 2 10 − 12 + 2 = (11 − 1)(1 + 5 + 5 ) = 0inZ31. These equations verify the Kirchhoff voltage law property for the voltage graph embedding of Fig. 4. This voltage graph has six edges (one for each generator in ) and three regions. There is only one vertex (after identiﬁcation of the two occurrences of each edge, matching up the arrows), as seen by the clockwise ordering of the generators and their inverses:

(−2, 10, − 5, − 6, − 10, − 12, 6, 1, 12, 2, − 1, 5).

The surface S in which the voltage graph K is embedded is orientable, since each edge appears once in each direction, giving all the clockwise (say) region 252 Arthur T. White

12 5

2 1 10 6

5 12

2 1

Fig. 4. A voltage graph for PG(2, 5)

boundaries. Euler’s formula yields h = 2, so S = S2. Then the covering surface has Euler characteristic 31ε(S2) =−62 = 2 − 2h,soh = 32 and the covering embedding is on S32. The thirty-one vertices in the covering embedding are the points of PG(2, 5). Let ρ denote the covering projection. Then the 31 lifts ρ−1(R) of the hexagon R depict the lines of PG(2, 5). Thus, if we start with the generator 1 on the boundary −1 of R, we ﬁnd that R0 = (0, 1, 3, 8, 18, 12), giving us l0 in ρ (R); similarly, Ri = R0 + i depicts li, for each i in Z31. In like manner, each triangular region below lifts to 31 triangular hyper-regions above. Clearly, Z31 acts as a group of automorphisms on the covering embedding. For example, adding i sends lj to lj+i. Similarly, Z31 acts regularly on the 31 hyper-regions covering each triangular region below. An (r, g)-cage is an r-regular graph of girth g with minimum order. Modifying the covering embedding, as described in Section 2, produces a hexagonal embedding of the Levi graph for PG(2, 5); this graph is a (6, 6)-cage, which we have now minimally embedded in S32 as well. From Biggs [2, Prop. 23.1(2)], we deduce the following result:

Theorem 3.3 The Levi graph G(PG(2), n) is an (n + 1, 6)-cage of order 2(n2 + n + 1).

The construction given above for the case n = 5 extends to all prime powers n ≡ 2 (mod 3), giving the following result.

Theorem 3.4 Let Gn be the (n + 1, 6)-cage that is the Levi graph for PG(2,n). 2 If n = 3m − 1 is a prime power, then γ(G3m−1) = m(3m − 2) .

Corollary If n = 3m−1 is a prime power,then γ(PG(2), 3m−1) = m(3m−2)2. 12 Embeddings and geometries 253

For prime powers n ≡ 0 or 1 (mod 3), similar analyses apply. A summary of the results is as follows: Theorem 3.5 Let n be a prime power. • If n ≡ 0 (mod 3), then PG(2,n)has a model on an orientable pseudosurface of characteristic (3 − 2n)(n2 + n + 1)/3 in which n2 + n − 1 of the hyper- regions are quadrilateral and all others are triangular; these embeddings are asymptotically efﬁcient, in terms of their Euler characteristic. • If n ≡ 1(mod 3), then PG(2,n)has a model on the orientable surface of genus 1+(n−1)(n2 +n+1)/3 in which n2 +n+1 of the hyper-regions are pentagonal and all others are triangular; the genus of these geometries is asymptotic to the genus of these surfaces. • If n ≡ 2(mod 3), then PG(2,n)has a model on the orientable surface of genus 1 + (n − 2)(n2 + n + 1)/3 in which all of the hyper-regions are triangular; thus these are minimum-genus embeddings for these geometries.

In each of the three cases, the group Zn2+n+1acts regularly on the point and line sets of PG(2, n), and also on each orbit of the set of hyper-regions. Moreover, the Euler characteristic is maximized in each case, subject to this last condition. We combine all three cases in the following result. Theorem 3.6 For i = 0, 1, 2 and prime power n ≡ 2 + i(mod 3), the maximum Euler characteristic among all orientable surfaces or pseudosurfaces on which ( n) PG 2, can be modelled so that Zn2+n+1 acts regularly on each orbit set of hyper-regions is (4 − 2n − i)(n2 + n + 1)/3. Figueroa-Centeno [5] generalized the approach illustrated here and found topological models for PG(m, n), when m + 1 is prime and n is a prime power, and where two distinct points belong to exactly λ lines for λ ≥ 2.

4. Afﬁne geometries

An affine plane = (P , L ) consists of a set of points P and a set of lines L , such that the following hold: ( 1) Any two distinct points of P lie on a unique common line. ( 2) Through a given point not on a given line, there is a unique parallel line (‘parallel’ means ‘disjoint’, as usual). ( 3) There exist four points, no three of which are collinear. Note that the first and third axioms agree with those for a projective plane, but that (2) has been replaced with Playfair’s axiom for the Euclidean plane (the parallel postulate, equivalent to Euclid’s fifth postulate). It is routine to show that 254 Arthur T. White commencing with a projective plane and removing one line (and all points on that line) produces an affine plane , and the process is reversible. For a projective plane = (n), we denote the corresponding affine plane by = (n). Thus we have the following consequences of Theorem 3.1.

Theorem 4.1 For a ﬁnite afﬁne plane (P , L ) of order n,

• |P |=n2; • |L |=n2 + n; • each point lies on exactly n + 1 lines; • each line contains exactly n points; • L can be partitioned into n+1 parallel classes, each class consisting of n lines.

Corollary Each afﬁne plane (n) is a resolvable balanced incomplete block design with v = n, b = n2 + n, r = n + 1, k = n and λ = 1.

In particular, if we start with (n) = PG(2,n), then we get (n) = AG(2,n), the classical afﬁne geometry of order n. In this section we consider topological models for these geometries (see White [17], [19] for details of the following constructions).

(1) The construction of AG(2, n) by deletions from PG(2, n) is modelled on the same topological space as the corresponding projective plane PG(2, n); we merely delete all the points on one line. However, this is unsatisfactory for at least two reasons: • Lines are no longer modelled by cyclic region boundaries, but now appear as paths, so the resulting loss of symmetry is displeasing, as are the suggestions of ‘betweenness’. • The embeddings are inefﬁcient: for example, AG(2, 2) inherits the torus from PG(2, 2), yet AG(2, 2) is planar; similarly, AG(2, 3) inherits the pseudosurface (S1;13(2)) from PG(2, 3), reducible to (S1;9(2)), but we can do much better (see Section 6). (2) For n = p, an odd prime, and ={(0, 1), (1, 0), (1, 1), ... , (1, p − 1)} for the group = Zp × Zp, the spherical voltage graph of Fig. 5 lifts to an embedding of the Cayley graph G() on Sh, the orientable surface of genus h = 1 + p(p − 2)(p + 1)/2. The p regions covering the loop carrying the generator (0, 1) are the coordinatized vertical lines in AG(2, p). The p regions covering the loop (1,m) are the lines of slope m, for 0 ≤ m ≤ p − 1. The shaded region in Fig. 5 is covered by p(p2 +p)-gons. Although this branched covering projection is not efﬁcient with respect to the Euler characteristic, the fact that it gives an explicit resolution of the p2 + p lines into p + 1 parallel classes of p lines each (each class partitioning the point set) is a nice feature of this model. 12 Embeddings and geometries 255

1 (p –1)

10 11

Fig. 5. A voltage graph for AG(2,p), p an odd prime

The construction of Fig. 5 does not extend to prime powers q = pm with exponent m>1, since the characteristic is still p. Thus the loops will lift to p-gons, whereas q-gons are required. (3) By grouping the loops of Fig. 5 into triples (as far as possible), with some voltages (1, m) changed to (c, cm) for an appropriate c in order to satisfy the Kirchhoff voltage law on the shaded region – that is, each boundary triangle, and perhaps one boundary pentagon, carries a voltage sum equal to the identity in Zp×Zp, and by repeated application of a construction ofWhite [20, Sec. 6c], we obtain models for AG(2, p) that are more efficient than those of either (1) or (2), while retaining the feature that each loop lifts to a class of parallel lines. For (b) and (c) below, the action of Zp × Zp on the set of hyper-regions is regular on each orbit, and the models found are optimally efficient (in terms of maximizing the Euler characteristic) for that feature. Note the similarity of this result to Theorem 3.7. (For AG(2, 3), see Section 6.) Theorem 4.2 Let n be an odd prime. (a) If n ≡ 0 (mod 3), the only possible affine geometry is AG(2, 3). (b) If n ≡ 1 (mod 3), AG(2, n) has a model on an orientable generalized pseudosurface of Euler characteristic −n(2n2 − 2n − 3)/3, in which n2 hyper-regions are pentagonal and all others are triangular. (c) If n ≡ 2 (mod 3), AG(2, n) has a model on an orientable generalized pseudosurface of Euler characteristic −n(2n2 − 4n − 3)/3, in which all hyper-regions are triangular. 256 Arthur T. White

5. 3-conﬁgurations

We have seen that the Fano plane is the ground case for the interesting collection PG(2, n) of finite geometries. It is also one of the smallest 3-configurations, and in this section we generalize the Fano plane in this second direction. A3-configuration is a finite geometry satisfying the following axioms: (CA1) Each line contains exactly three points. (CA2) Each point lies on exactly r lines, for some fixed positive integer r. (CA3) Any two distinct points lie on at most one common line. If we want to specify the value of r, we call the geometry a (3, r)-configuration. We could generalize further to k-configurations for arbitrary k ≥ 3 (since a 2-configuration is just an r-regular graph), but here we focus on k = 3 for three reasons: • If k ≥ 4, then chordal points on the boundary of a line-region are collinear, but they might not be adjacent in the graph being embedded; for k = 3, the embedded graph is automatically the Menger graph. • Many well-known geometries are 3-configurations, as we shall see; thus, there is plenty of interest in this natural value of k. • Only for k = 3 are there no connotations of ‘betweenness’ in our models of lines as region boundaries. A3-configuration C is a (v, b, r,3;0,1)-design, where two non-adjacent vertices in the Menger graph M(C) represent points with no common line (λ1 = 0), and two adjacent vertices in M(C) represent points with exactly one line in common (λ2 = 1).IfM(C) is complete, then the design is a balanced incomplete block design (BIBD), while if M(C) is strongly regular, then the design is a partially balanced incomplete block design (PBIBD). The central idea of this section is that 3-configurations correspond to K3- decompositions of their Menger graphs, and hence they can be modelled by bichromatic-dual embeddings of these graphs for which one colour class consists of triangular regions depicting the lines of the geometry. We call such an embedding a configuration embedding of degree r, where r is the replication number, as in axiom (CA2). The toroidal embedding in Fig. 2 of the Menger graph K7 for the Fano plane illustrates this concept for r = 3. The following theorem appears in White [21]. Theorem 5.1 Let G be a 2r-regular graph of order v. Then the following are equivalent: • G is the Menger graph for a (3, r)-configuration; • K 1 vr Gis 3-decomposable into 3 3-cycles; • G has a configuration embedding of degree r. 12 Embeddings and geometries 257

The next theorem is due to Gropp [8].

Theorem 5.2 There exists a (3,r)-conﬁguration on v points if and only if v ≥ 2r + 1 and 3 divides vr.

Let ε(C) denote the maximum Euler characteristic, taken over all topological models, of the 3-conﬁguration C. The following theorem of White [20] is surprisingly general.

Theorem 5.3 Let C = (P , L) be a 3-conﬁguration, with parameters (v, b, r, 3: 0, 1). Then ε(C) = v − b.

Proof The maximum Euler characteristic for C is attained by a cellular embedding (on some surface, pseudosurface, or generalized pseudosurface) of the Menger graph M(C). This graph has v vertices, vr = 3b edges, b line-regions, and at most b hyper-regions (with exactly b if all hyper-regions are triangular). We can always embed M(C) as bK3 on bS0 with appropriate vertex identiﬁcations, so by Euler’s formula, ε(C) = v − b.

We check that, for the generalized pseudosurface T = (bS0; v(r)) just constructed,

ε(T ) = 2b − v(r − 1) = 2b − 3b + v = v − b.

Thus, the characteristic of C = (P, L) depends only on the sizes of P and L. However, as we generally prefer more elegant models than those constructed in the proof, our search is not ended. In White [21], we find the following criteria for heightened interest. • The Menger graph should be taken, wherever possible, as a Cayley graph G(), where || = v; this facilitates the construction of a configuration embedding (using an appropriate voltage graph) and is relevant to the next criterion (see White [19, Theorems 16–24]). • We seek to maximize the order of the automorphism group of the map (since symmetries are pleasing). • We prefer the block design to be balanced (that is, λ = 1, which requires M(C) to be Kv) or partially balanced (that is, λ1 = 0 and λ2 = 1, which requires M(C) to be strongly regular). • We prefer the embedding space T to be a proper surface. • We prefer orientable to non-orientable embeddings. • We seek to maximize the Euler characteristic of T . • We especially like to model previously studied geometries in this new context. • We find some pleasure in using unusual groups. 258 Arthur T. White

Models have been found for all 320 pairs (v, r) satisfying vr ≡ 0 (mod 3) and 2r + 1 ≤ v ≤ 50 (see Theorem 5.2). In White [21], some of the more interesting geometries among the 320 (according to the above criteria) are discussed briefly. In this and the following section, we select for discussion some particular small-order 3-configurations (using the penultimate criterion), and then apply the remaining criteria to construct our models. The Möbius–Kantor geometry is an (8, 8, 3, 3; 0, 1)-PBIBD and is modelled by interpreting the voltage graph embedding of Fig. 2 (previously used with Z7 for K7) for the voltage group Z8; this embeds the Menger graph K4(2) on S1 and shows that the Möbius–Kantor geometry joins the Fano plane in having genus 1. There are several theorems in Euclidean geometry that establish concurrency for triples of lines associated with a triangle. For example, the three perpendicular bisectors meet at the circumcentre; the three altitudes meet at the orthocentre; the three internal angle bisectors meet at the incentre; and the three medians meet at the centroid. If the triangle is equilateral, then all four points agree and the model of the Fano plane in Fig. 1 depicts this (and the incircle as well). Similarly, two additional theorems from plane geometry give rise to familiar finite geometries. We model the geometry resulting from a theorem of Pappus in the next section, but here we recall a theorem of Desargues from projective geometry: two triangles in perspective from a point are also in perspective from a line (see Fig. 6). The corresponding Desargues geometry is a (10, 10, 3, 3; 0, 1)-PBIBD, and its Menger graph M is the complement of the Petersen graph. But since the Petersen graph is not a Cayley graph, neither is M, and thus there is no index-1 voltage graph embedding available.

E B A C F

IJ H

Fig. 6. A Desargues conﬁguration 12 Embeddings and geometries 259

E B

III F II JC

HI I J I I GD

J H

D BAFE III II

C HE C IG

II III F G BD J H

Fig. 7. A model of the Desargues geometry on S2

In Fig. 7 we present an embedding of M in S2, found by Figueroa-Centeno and White [6] by ad hoc means. The line region I and the hyper-regions II and III appear three times each, thus displaying the 3-fold rotational symmetry α = (A)(BCD)(EF G)(H IJ ) that generates the map-automorphism group Aut(Map) = α ≡ Z3. The orbits of the action of this group on the point set of the Desargues geometry are the pole A, the polar points H , I and J , and the triangles of perspectivity BCD and EFG. The induced action on the line set has as its orbits the three lines containing the pole A, the three lines bounding the triangle BCD, the three lines bounding the triangle EFG, and the polar line HIJ. Note that Aut(Map) ﬁxes the point of perspectivity, the line of perspectivity, and the two triangles of perspectivity, but nothing else. The usual current graph used to embed K19 on its genus surface S20 (see White [19, Fig. 9-6]) is the triangular prism K2 C3. Since this graph is not bipartite, the 260 Arthur T. White regions of the dual of the covering embedding cannot be 2-coloured — but using K3,3 instead (the other cubic graph of order 6) remedies this defect. This produces two non-isomorphic triangulations by K19 on the same surface (see Lawrencenko, Negami and White [12]). The latter embedding yields two Steiner triple systems (one from each colour class) and one Mendelsohn triple system (by taking every region as a block) of order 19. The voltage group Z3 × Z3 × Z3 can be used to embed K27 minimally in S46, with bichromatic dual. The resulting Steiner triple system is resolvable, and hence forms a Kirkman triple system.

6. Partial geometries

In this section we consider 3-conﬁgurations that are also partial geometries. Historically, the word ‘partial’indicated that some pairs of points are not collinear. The added axiom to those for a 3-conﬁguration is this:

(CA4) There is a non-negative integer α so that, for each line l and each point p not on l, l contains exactly α points that are collinear with p.

Thus, for a partial geometry 3-configuration, the possible values for α are 0, 1, 2 and 3. We begin with four geometries, each on nine points, with one for each value of α. We use the spherical voltage-graph embedding of Fig. 5, specialized to p = 3 and voltage group Z3 × Z3. α = 3. As we saw before, the covering embedding of Fig. 5, for p = 3, models the 3-configuration AG(2, 3) on S7. Because the Menger graph is K9, it is easy to check that α = 3. For a more efficient model ofAG(2, 3), we can use the projective plane voltage graph of Fig. 8, again with voltage group Z3 × Z3; in this case, the surface is N5. Both models for AG(2, 3) yield, as expected, a (9, 12, 4, 3, 1)-BIBD, a Steiner triple system with 9 points. α = 2. From Fig. 5, specialized to p = 3, remove the loop carrying the voltage 12 = (1, 2). The generating set is reduced to {(0, 1), (1, 0), (1, 1)}, the Kirchhoff voltage law is satisfied, and the covering embeds the strongly regular complete tripartite graph K3(3) into its genus surface S1. Fig. 9 shows the resulting Pappus geometry, coordinatized so as to emphasize its containment in AG(2, 3). From the model, it is routine to check that α = 2, and to see that the Pappus geometry contains three of the four parallel classes of lines from AG(2, 3). The resulting design is a (9, 9, 3, 3; 0, 1)-PBIBD. We note that this geometry arises from Pappus’s theorem in Euclidean geometry (see Fig. 10): if A, B, C are three points on a line l and if A ,B ,C are three points on another line l , then the 12 Embeddings and geometries 261

11 12 10 10

Fig. 8. A voltage graph for AG(2, 3)

y = 0 12 02 00 20 21 11 x = 0 x = 2

10y = x + 2 01y = x + 1 22y = x 10

y = 1 y = 2 20 21 11 12 02 00 x = 1

Fig. 9. A model of the Pappus conﬁguration on S1

A B 0 C D E

0 F A B

C 0

Fig. 10. A Pappus conﬁguration 262 Arthur T. White intersection points of the three pairs of lines AB and A B, AC and A C, and BC and B C are collinear.

α = 1. Now further remove the loop in Fig. 5 carrying the voltage 11 = (1, 1), still specialized to p = 3. The new generating set is {(1, 0), (0, 1)} and the covering embedding is of the strongly regular graph C3 C3 into S1, where all hyper-regions are hexagonal. From AG(2, 3) only the vertical and horizontal parallel classes now survive, and we can easily see that α = 1. The design is a (9, 6, 2, 3; 0, 1)-PBIBD.

α = 0. Finally, remove one of the two remaining loops. The single loop now lifts (still using Z3 × Z3) to an embedding of the strongly regular graph 3K3 on S0 (three copies of K3 on S0), giving a 3-conﬁguration with α = 0 and a (9, 3, 1, 3; 0, 1)-PBIBD.

We now consider generalizations of these four constructions.

α = 0. The strongly regular graph nK3 on nS0 gives a (3n, n,1, 3; 0, 1)-PBIBD. Since r = 1, it is immediate that α = 0.

α = 1. A partial geometry with α = 1 is called a generalized quadrangle. Only three 3-conﬁgurations are generalized quadrangles (see Cameron [3]):

(1) The smallest has order 9, and is discussed above. (2) The points are the ﬁfteen duads xy, where 1 ≤ x

α = 2. Here we take the Menger graph K3(n) (the complement of the Menger graph 2 nK3 used for α = 0) to model a geometry that is a (3n, n , n, 3; 0, 1)-PBIBD. When n = 2, the result gives the Pasch conﬁguration. When n = 3, we obtain Pappus’s geometry, using Z3 × Z3 as in Fig. 9. Alternatively, instead of lifting 12 Embeddings and geometries 263 the 1-vertex, 3-loop voltage graph from the sphere, we can again use the toroidal voltage-graph embedding of Fig. 2 with this third voltage group: Z7 for the Fano plane, Z8 for the Möbius–Kantor geometry, and now Z3 ×Z3 (with suitable change of generators) for the Pappus geometry.

0 1 n – 1

Fig. 11. A voltage graph for Kn,n

The general construction (see Stahl and White [16]) begins with the voltage- graph embedding (the dipole Dn on n edges in S0) shown in Fig. 11, using Zn as K S h = 1 (n− )(n− ) voltage group. The covering embedding is n,n in h, where 2 1 2 , and has n regions, each bounded by a Hamiltonian cycle. For instance, the shaded region in Fig. 11 lifts to a 2n-gon with clockwise region boundary (a0, b1, a1, b2, a2, b3, ..., a(n − 1), b0). Adding n new vertices, one in the interior of each region, and joining each new vertex to every vertex in the Hamiltonian boundary, we obtain a triangular (and hence genus) embedding of K3(n) in Sh. (Note that we γ(K ) = 1 (n − )(n − ) have just proved that 3(n) 2 1 2 .) It is a nice exercise to show that the dual of this embedding is bichromatic (in fact, every triangular embedding of K3(n) has this property), and that the geometry obtained by taking all the regions of either colour as lines produces a 3-conﬁguration that is a partial geometry with α = 2, for each n.

α = 3. A bichromatic-dual triangulation of an orientable surface exists for Kn precisely when n ≡ 3 or 7 (mod 12) (see Ringel [13] and Youngs [22], (n, 1 n(n − ), 1 (n − ) ) respectively). By an easy argument, these 6 1 2 1 ,3,1-BIBDs (or STS(n) designs) have α = 3 uniformly. Similarly, each Steiner triple system of any order v ≡ 1 or 3 (mod 6) has α = 3, and can be modelled in a variety of ways, although not necessarily with all hyper-regions being triangular. 264 Arthur T. White

7. Regular embeddings for PG(2, n)

An embedding of the bipartite incidence graph (the Levi graph) of PG(2, n) into an orientable surface is regular if its automorphism group induces a regular action on the set of all ordered pairs (p, l) (called ﬂags), where the point p belongs to the line l. This occurs precisely when n = 2 and 8, in agreement with the work of Higman and McLaughlin [10] in an algebraic setting, and of Singerman [15] for planes represented as hypermaps. There are ﬁve regular embeddings of the corresponding Menger graphs in total, and each can be constructed using an index-1 voltage graph embedding (see Fink and White [7]). We can verify the regularity condition by showing that (Aut M)0 is generated by multiplication by some number b with n + | M| = (n2 + n + )(n + ) order 1inZn2+n+1, since then Aut 1 1 . n = 2. The Fano plane is modelled on both S1, as shown in Fig. 2 (with b = 2), and on S3, covering the Z7 voltage graph embedding of Fig. 12 (again with b = 2).

1 4

Fig. 12. Another voltage graph for K7 and the Fano plane n = 8. PG(2, 8) is modelled by using ={1, 2, 4, 8, 16, 32, 64, 55, 37} for = Z73; the covering Menger graph is then G(). In each case, the embedding below commences with an unshaded nonagon having a clockwise ordering of generators, in increasing powers of 2, as recorded above. The 73 lifts of this nonagon, (i, i + 1, i + 3, ... , i + 36), one for each i in Z73, represent the lines of PG(2, 8): this is because each is a translate of a planar cyclic difference set for Z 73. Since the integer 9 has three uniform partitions (1 × 9, 9 × 1 and 3 × 3), there are three ways to complete a voltage graph embedding for PG(2, 8) – add one nonagon, nine loops, or three triangles – for maximal symmetry. The ﬁrst two of these are analogous to those for n = 2 above, so we omit ﬁgures for them. • Attach a second (but shaded) nonagon, with a clockwise ordering of generators (−1, −2, −4, ..., −37) along any one edge (the edge carrying voltage 4, for example). The covering embedding has 73 nonagons as hyper-regions and is on the surface S220; b = 4. • Attach nine loops to the initial nonagon, making a bouquet of nine circles, similar to the bouquet of three circles for n = 2. The covering embedding has nine 73-gons as hyper-regions and is on the surface S252; b = 2. 12 Embeddings and geometries 265

• The voltage graph embedding in S3 is given in Fig. 13. The covering embedding is into the surface S147, with 3 × 73 = 219 triangular hyper-regions; b = 4. This is the embedding that shows that γ (PG(2, 8)) = 147 (see Corollary 3.6).

16 55 8 37

32 2 61 1 4

37 8

55 16

64 32

Fig. 13. A voltage graph for PG(2, 8)

8. Problems

We conclude with some problems for further study. Models for AG(2, 3). Models for this geometry have been found on the following topological spaces: (a) N5; (b) S(1; 3(2)); (c) (S1 ∪ 3S0;9(2)); (d) S3; (e) S4; (f) S7; (g) S((1);9(2)). The ﬁrst three maximize the Euler characteristic, but only (a) is a surface, and it is non-orientable. No orientable surface triangulation is possible, but (b) gives an orientable pseudosurface triangulation. Model (c) starts with Pappus’s geometry and then adds a fourth parallel class of lines. Models (d) and (e) are constructed by rotation system and voltage graph lifting, respectively, with the former being the most efﬁcient orientable surface model possible. Model (f) has the feature that the resolvability of the block design is explicit, and model (g) is the result of deletions from PG(2, 3). Thus, each of the seven models has some distinguishing feature, yet each is defective in some way. So it is natural to ask which model – one of these, or perhaps some other – is the most satisfying model for AG(2, 3)?

Models for AG(2, n). In general, the models forAG(2, n) obtained by deletion from PG(2, n) are not very satisfying. Furthermore, while the surface models for prime 266 Arthur T. White values of n given in Section 4 make explicit the resolvability of the corresponding block design, they are not very efﬁcient. Can more pleasing models be found, say when n is a prime power?

Specific 3-configurations. In Section 5, we considered models for some 3- configurations of low order. Among those with order at most 50, the two with (v, r) = (36, 15) and (42, 18) stand out in that, as modelled, they do not correspond to PBIBDs since their Menger graphs are not strongly regular. It would be satisfying to construct models for geometries on these parameters, with the strongly regular graphs K6(6) and K7(6) as Menger graphs. In other words, are these graphs K3-decomposable, and if so, can ‘nice’ configuration embeddings be found?

Embeddings for line graphs L(Kn). Our model for the Pasch conﬁguration, given in Section 6, has L(K4) (that is, the octahedal graph K3(2)) as its Menger graph, and our model for the Desargues geometry in Section 5 has L(K5) as its Menger graph. (Also, L(K3) is trivially the Menger graph for K3 on the sphere.) In general, L(Kn) has a K3-decomposition (one can be obtained from the set of all 3-cycles in Kn), so any geometry with L(Kn) as its Menger graph corresponds to a PBIBD ( 1 n(n − ) 1 n(n − )(n − ) n − with parameters 2 1 , 6 1 2 , 2, 3; 0, 1). Hence we ask: what interesting embeddings can be found for L(Kn), n ≥ 6?

Generalized quadrangle 3-conﬁgurations. There are two speciﬁc problems here:

• Determine whether the generalized quadrangle 3-configuration of order 15 has genus 3 or 4. (In Section 6, we gave a configuration embedding for this geometry on S4, and it is easy to show that at least three handles are needed.) • Find a pleasing surface model for the generalized quadrangle 3-configuration of order 27.

Embeddings for L(Kn). The embedding for the generalized quadrangle 3-conﬁguration mentioned in the previous problem has L(K6) as its Menger graph. For n ≡ 0 or 1 (mod 4), L(Kn) is regular of odd degree, so no K3-decomposition is possible. What happens when n ≡ 2 or 3 (mod 4) for n ≥ 7? Such a decomposition ( 1 n(n− ) 1 n(n− )(n− )(n− ) 1 (n− )(n− ) ) would yield a 2 1 , 24 1 2 3 , 4 2 3 ,3;0,1 -PBIBD. Can interesting models be found?

References

1. I. Anderson, Combinatorial Designs: Construction Methods, Horwood, 1990. 2. N. Biggs, Algebraic Graph Theory (2nd edn.), Cambridge University Press, 1993. 3. P. J. Cameron, Finite geometries, Handbook of Combinatorics (eds. R. Graham, M. Grötschel and L. Lovász), Elsevier Science (1995), 647–691. 12 Embeddings and geometries 267

4. H. S. M. Coxeter, Self-dual configurations and regular graphs, Bull. Amer. Math. Soc. 56 (1950), 413–455. 5. R. M. Figueroa-Centeno, Surface Models of Finite Geometries, Ph.D. thesis, Western Michigan University, 1998. 6. R. M. Figueroa-Centeno and A. T. White, Topological models for classical configurations, J. Statist. Planning and Inference 86 (2000), 421–434. 7. J. B. Fink andA. T.White, Regular imbeddings of finite projective planes, Graph Theory, Combinatorics, and Applications, Vol. 2 (eds. Y.Alavi and A. Schwenk), Wiley (1995), 1233–1241. 8. H. Gropp, Nonsymmetrical configurations with natural index, Discrete Math. 124 (1994), 87–98. 9. J. L. Gross, Voltage graphs, Discrete Math. 18 (1974), 239–246. 10. J. L. Gross and T. W. Tucker, Generating all graph coverings by permutation voltage assignments, Discrete Math. 18 (1977), 273–283. 11. D. G. Higman and J. E. McLaughlin, Geometric ABA-groups, Illinois J. Math. 5 (1961), 382–397. 12. S. Lawrencenko, S. Negami and A. T. White, Three nonisomorphic triangulations of an orientable surface with the same graph, Discrete Math. 135 (1994), 367–369. 13. G. Ringel, Map Color Theorem, Springer, 1974. 14. J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), 377–385. 15. D. Singerman, Klein’s Riemann surface of genus 3 and regular imbeddings of finite projective planes, Bull. London Math. Soc. 18 (1986), 364–370. 16. S. Stahl and A. T. White, Genus embeddings for some complete tripartite graphs, Discrete Math. 14 (1976), 279–296. 17. T. R. Walsh, Hypermaps versus bipartite maps, J. Combin. Theory (B) 18 (1975), 155–163. 18. A. T. White, Efficient imbeddings of finite projective planes, Proc. London Math. Soc. 70 (1995), 33–55. 19. A. T. White, Graphs of Groups on Surfaces: Interactions and Models, North-Holland, 2001. 20. A. T. White, Modelling finite geometries on surfaces, Discrete Math. 244 (2002), 479–493. 21. A. T. White, Modelling 3-configurations on surfaces, Electron. Notes Discrete Math. 11, Elsevier, 2002. 22. J. W. T. Youngs, The mystery of the Heawood conjecture, Graph Theory and its Applications (ed. B. Harris), Academic Press (1970), 17–50. 13 Embeddings and designs

M. J. GRANNELL and T. S. GRIGGS

1. Introduction 2. Steiner triple systems and triangulations 3. Recursive constructions 4. Small systems 5. Cyclic embeddings 6. Concluding remarks References

When a graph is embedded in a surface, the faces that result can be regarded as the blocks of a combinatorial design. The resulting design may be thought of as being embedded in the surface. This perspective leads naturally to a number of fascinating questions about embeddings, in particular about embeddings of Steiner triple systems and related designs. Can every Steiner triple system be embedded, can every pair of Steiner triple systems be biembedded, and how many embeddings are there of a given type?

1. Introduction

In this section we deﬁne the terminology taken from combinatorial design theory and summarize some of the basic results. Before doing this, we remark that the study of the relationship between block designs and graph embeddings dates back to Heffter, who in 1891 realized the connection between two-fold triple systems and surface triangulations. Later work in this ﬁeld was done by Emch [27], Alpert [2], White [57], Anderson and White [6], Anderson [4], [5], Jungerman, Stahl and White [40], Rahn [54], and more recently White [58]. These authors considered various aspects of the above relationship, including embeddings into closed surfaces, pseudosurfaces and generalized pseudosurfaces, embeddings of

268 13 Embeddings and designs 269 balanced incomplete block designs (BIBDs) with block size greater than 3, and the symmetry properties of the resulting embeddings. However, the material we survey in this chapter is mainly, although not exclusively, concerned with embeddings of Steiner triple systems in both orientable and non-orientable surfaces. Embeddings in pseudosurfaces and generalized pseudosurfaces will not be considered. The reader is assumed to be familiar with embeddings of graphs in both orientable and non-orientable surfaces, and the description of such embeddings by means of rotation schemes. Some familiarity with the use of current and voltage graphs in the construction of embeddings is assumed (see Chapter 1). When referring to the number of embeddings (or other combinatorial objects), we mean the number of non-isomorphic embeddings (or objects) of the specified type. In order to reduce the number of references, particularly for results in design theory, we give as secondary sources the books Triple Systems by Colbourn and Rosa [21] and The CRC Handbook of Combinatorial Designs, edited by Colbourn and Dinitz [20]. The principal item required from design theory is the following definition. A Steiner triple system of order n is a pair (V, B), where V is an n-element set (the points) and B is a collection of 3-element subsets (the blocks)ofV such that each 2-element subset of V is contained in ‘exactly one block’ of B. It is well known that a Steiner triple system of order n (briefly STS(n)) exists if and only if n ≡ 1 or 3 (mod 6). If, in the definition, the words ‘exactly one block’ are replaced by ‘exactly two blocks’, then we have a two-fold triple system of order n, TTS(n) for short. A two-fold triple system of order n exists if and only if n ≡ 0or1 (mod 3). If a TTS(n) has no repeated blocks, it is said to be simple. A (possibly non-simple) TTS(n) may be obtained by combining the block sets of two STS(n)s that have a common point set. An STS(n) can be considered as a decomposition of the complete graph Kn into triangles (copies of K3); likewise a TTS(n) can be considered as a decomposition of the two-fold complete graph 2Kn (in which there are two edges between each pair of vertices) into triangles. Up to isomorphism, there is just one STS(n) for n = 3, 7, 9, while there are two for n = 13, precisely one of which is cyclic (that is, has an automorphism of order 13). There are 80 STS(15)s, of which two are cyclic, and there are 11 084 874 829 STS(19)s (see [41]), of which four are cyclic. The number of non-isomorphic n2( 1 +o(1)) STS(n)s is n 6 as n →∞ [59], and, again speaking asymptotically, almost all of these have only the trivial automorphism group [7]. A transversal design of order n and block size 3 is a triple (V, G, B), where V is a3n-element set (the points), G is a partition of V into 3 parts (the groups) each of size n, and B is a collection of 3-element subsets (the blocks)ofV such that each 2-element subset of V is either contained in exactly one block of B or in exactly one group of G, but not both. A transversal design of order n and block size 3 is 270 M. J. Grannell and T. S. Griggs denoted by TD(3,n); since we consider only block size 3, we will simply speak of a transversal design of order n.ATD(3,n)may be considered as a decomposition of a complete tripartite graph Kn,n,n into triangles, with the tripartition defining the groups of the design. A TD(3,n) is equivalent to a Latin square of side n in which the triples are given by (row, column, entry). AMendelsohn triple system of order n is defined in a similar fashion to an STS(n) except that triples and pairs are taken to be ordered, so that the cyclically ordered triple (a,b,c)‘contains’ the ordered pairs (a, b), (b, c) and (c, a). A Mendelsohn triple system of order n, MTS(n) for short, exists if and only if n ≡ 0 or 1 (mod 3) and n = 6. An MTS(n) may be considered as a decomposition of a complete directed graph on n vertices into directed 3-cycles. If the directions are ignored, then an MTS(n) gives a TTS(n). To see the connection between design theory and graph embeddings, consider the case of an embedding of a complete graph Kn in an orientable surface in which all the faces are triangles. Taking these triangles with a consistent orientation to form a set of blocks, the faces of the embedding yield a Mendelsohn triple system of order n. Similarly, a triangulation of Kn in a non-orientable surface gives a two-fold triple system of order n. The precise correspondence between such systems and triangulations is given in [48] and involves pseudosurfaces. Our interest is in the question: which Mendelsohn (respectively, two-fold) triple systems occur as triangulations of Kn in orientable (respectively, non-orientable) surfaces? The following answer is given in [24]. Let (V, B) be a TTS(n). For each x ∈ V , define a neighbourhood graph Gx: its vertex-set is V \{x}, and two vertices y and z are joined by an edge if {x,y,z}∈B. Clearly, Gx is a union of disjoint cycles. A TTS(n) occurs as a triangulation of a non-orientable surface if and only if every neighbourhood graph consists of a single cycle. If the blocks of the TTS(n) can be ordered to form an MTS(n), then the surface is orientable. We now move on to the much more interesting relationship between embeddings and Steiner triple systems.

2. Steiner triple systems and triangulations

Let (V, B) be an STS(n) and let Kn be the complete graph with vertex-set V . By an embedding of (V, B) in a surface S (which may be orientable or non- orientable) we mean any embedding φ: Kn → S with the property that for each {u, v, w}∈B, the 3-cycle (uvw) constitutes the boundary of some face of φ. For the sake of convenience, we abbreviate the above deﬁnition by just saying that in the embedding φ, every triple of B is facial. Since each edge of Kn belongs to precisely one facial triple, the faces of φ can be properly 2-coloured. Usually we colour the facial triples of B black and the remaining faces white. 13 Embeddings and designs 271

Conversely, let ψ: Kn → S be an embedding whose faces can be properly 2-coloured (black and white) with all black faces bounded by 3-cycles. Then ψ is an embedding of some STS(n). Indeed, let B be the collection of the 3-subsets of V = V(Kn) that correspond to the boundary triangles of black faces. Since our face colouring is proper, no edge is on the boundary of just one face. Thus, each edge of Kn is incident to precisely one black face, and so each pair of elements of V belongs to precisely one 3-subset of B. Hence (V, B) is an STS(n). A particularly interesting case occurs when the family of all white faces constitutes an STS as well. Let (V, B) and (V, B) be two STSs on a common point set V with |V |=n. We say that the pair {B, B} is biembeddable (or that the two STSs are biembeddable) in some surface S if there is an embedding φ of (V, B) whose white faces are 3-cycles constituting the blocks of an STS isomorphic to (V, B). In such circumstances, φ is called a biembedding. Brieﬂy, in a biembedding φ of the pair {B, B}, the facial triples of B are black while those corresponding to B are white. Necessarily, the biembedding φ is then a triangular embedding of a complete graph on n vertices and the surface has minimum genus. Conversely, each triangular embedding ψ: Kn → S whose faces can be properly 2-coloured induces a biembedding of a pair of STSs. In the orientable case we must then have n ≡ 0, 3, 4 or 7 (mod 12) (see Chapter 1) and combining this with the existence condition for STSs, we see that a pair of STSs on n points can have an orientable biembedding only if n ≡ 3 or 7 (mod 12).Asimilar argument in the non-orientable case shows that we must then have n ≡ 1 or 3 (mod 6). We illustrate these concepts in Fig. 1, which depicts a biembedding of a pair of isomorphic STS(7)s. Speciﬁcally, B ={013, 124, 235, 346, 450, 561, 602} and B ={023, 134, 245, 356, 460, 501, 612}. Because of the connection between biembeddings of STSs and face 2-colourable triangular embeddings of complete graphs, we recall a few facts

0 0

2 4 6 1 3 5

0 0

Fig. 1. A biembedding of STS(7)s in the torus 272 M. J. Grannell and T. S. Griggs about the latter. Constructions of minimum genus embeddings (which include triangulations) of complete graphs in orientable and non-orientable surfaces have a rich history. They form the essential part of the solution of the famous Heawood problem of determining the chromatic number of a surface, or, equivalently, determining the genus of a complete graph. Most of the solution (which also gave birth to modern topological graph theory as treated in [38]) is due to Ringel and Youngs; we recommend Chapter 1 or Ringel’s book [56] for details. However, the majority of the known minimum genus embeddings of complete graphs are not face 2-colourable. In the case when n ≡ 3 (mod 12), the orientable embeddings of Kn found in [56] are indeed face 2-colourable. The proof technique there uses the theory of current graphs. However, no information is yielded concerning the STS(n)s that have been biembedded. Below, in Theorem 2.1, we outline a proof of this result using exclusively design-theoretic methods. To our mind this is not only simpler and more transparent, but it also positively identiﬁes the STS(n)s so embedded. They are those obtained from the well-known Bose construction based on a Latin square constructed as the square-root Cayley table of an odd-order cyclic group [18].

Theorem 2.1 If n ≡ 3 (mod 12), then there exists a pair of biembedded Steiner triple systems of order n in some orientable surface.

Outline of Proof Take the additive group Z4s+1 and deﬁne on it the operation ◦ by i ◦ j = (i + j)/2 = (2s + 1)(i + j). Use the classical Bose construction [18] to build an STS (V, B) on the point set V = Z4s+1 × Z3. The block set B consists of 4s + 1 triples of the form (i, 0), (i, 1), (i, 2), i ∈ Z4s+1, together with 3 × (4s + 1)2s triples of the form (i, k), (j, k), (i ◦ j,k + 1), where i, j ∈ Z4s+1, i = j and k ∈ Z3. Let n = 12s + 3. We deﬁne two STSs (Zn, B0) and (Zn, B1), both isomorphic to (V, B), using the bijections fm: V → Zn, m = 0, 1, given by fm(i, k) = m 3i + (−1) kt where t = 6s + 1; naturally, Bm = fm(B). On the right side of the equation for fm(i, k) we have i ∈{0, 1,...,4s}, k ∈{0, 1, 2}, and the addition is modulo n. It can easily be checked that the two STSs are disjoint. It remains only to check that the pair {B0, B1} is biembeddable in an orientable surface. This is routine but somewhat tedious, and we refer the reader to the original paper [35].

We note here that a similar approach (that is, constructing triangular embeddings of Kn using the Bose construction) can also be found in [23]. However, the proof given there, which applies to all n ≡ 3 (mod 6), n ≥ 9, always produces an embedding in a non-orientable surface. In the case n ≡ 7 (mod 12), there are the toroidal embedding of K7 given above and the face 2-colourable triangular embedding of K19 given in [57] (see also [49]). 13 Embeddings and designs 273

Youngs [61] produced orientable triangular embeddings of Kn by means of current assignments on ladder graphs. Amongst the variety of ladder graphs used in [61], it is possible to ﬁnd, for each n ≡ 7 (mod 12), one which is bipartite [61, pp. 39–44]. Anderson [5] points out the signiﬁcance of a bipartition; for our purposes this ensures that the corresponding triangular embedding is face 2-colourable. Thus it is known that there are orientable biembeddings for all n ≡ 7 (mod 12). But here also, no information is produced about the STSs that have been embedded. In Section 3 we describe a recursive construction, a topological analogue of a well-known design-theoretic construction, which shows that such embeddings exist for half of the residue class n ≡ 7 (mod 12). This particular method has the additional advantage that it produces a large number of new embeddings, as will also be described in the next section.

3. Recursive constructions

The recursive construction that appears in Theorems 1 and 2 of [34] in a topological form, and again in Theorems 2 and 3 of [35] in a design-theoretical form, takes a biembedding of two STS(n)s and produces a biembedding of two STS(3n − 2)s. Here we give an informal description of this construction and then discuss extensions and related constructions. The construction commences with a given biembedding of two STS(n)s, which is equivalent to a face 2-colourable triangulation of Kn. We ﬁx a particular vertex ∗ ∗ ∗ z of Kn and, from the embedding, we delete z , all edges incident with z and all the triangular faces incident with z∗. The resulting surface S now has a hole ∗ ∼ whose boundary is an oriented Hamiltonian cycle in G = Kn − z = Kn−1.We next take three disjoint copies of the surface S, all with the same colouring and, in the orientable case, the same orientation; we denote these by S0,S1 and S2, and use superscripts in a similar way to identify corresponding points on the three surfaces. For each white triangular face (uvw) of S, we ‘bridge’ S0,S1 and S2 by gluing a torus to the three triangles (uiviwi) in the following manner. We take a face 2-colourable triangulation in a torus of the complete tripartite graph K3,3,3 with the three vertex parts {ui}, {vi} and {wi} and with black faces (uiwivi), for i = 0, 1, 2 (see Fig. 2). In the orientable case, the orientation of the torus must induce the opposite cyclic permutation of {ui,vi,wi} to that induced by the surfaces Si; this is important for the integrity of the gluing operation where black faces (uiwivi) on the torus are glued to the white faces (uiviwi) on S0,S1 and S2, respectively. After all the white triangles have been bridged we are left with a new connected triangulated surface with a boundary. We denote this surface by . It has 3n − 3 vertices and the boundary comprises three disjoint cycles, each of length n − 1. In 274 M. J. Grannell and T. S. Griggs

u0 v2 w0 u0

w2 u2 v1 v1

u1 v0 w1 w1

u0 v2 w0 u0

Fig. 2. Toroidal embedding of K3,3,3

order to complete the construction to obtain a face 2-colourable triangulation of K3n−2 (which gives a biembedding of two STS(3n − 2)s), we must construct an auxiliary triangulated bordered surface S and paste it to so that the three holes of are capped. To do this, suppose that D = (u1u2 ...un−1) is our oriented ∗ Hamiltonian cycle in G = Kn − z . Since n is odd, each alternate edge of D is incident with a white triangle in S; let these edges be u2u3,u4u5,...,un−1u1. i The surface S has, as vertices, the points uj for i = 0, 1, 2 and j = 1, 2,..., n − 1 together with one additional point, which we call ∞. Suppose initially that n ≡ 3 (mod 12). We may then construct S from the oriented triangles listed opposite in Table 1. The reason for the classification of the triangles into types 1 and 2 will become apparent shortly. Precisely how S is constructed is described in more detail in [34], where it is also proved that the final graph that triangulates the final surface is indeed K3n−2. The importance of the condition n ≡ 3 (mod 12) is that it ensures that the resulting surface is a closed surface and not a pseudosurface. Equivalently, it ensures that the neighbourhood graph of the point ∞ comprises a single cycle of 3n−3 points rather than a union of shorter cycles.As given above, the construction does not work for n ≡ 7 (mod 12); however we can modify it by taking a single value of j ∈{1, 3, 5,...,n− 2} and applying a ‘twist’ to the type-1 triangles associated with this value of j. To do this, we replace them by those shown in Table 2. Again, for an explanation of why this works, see [34]. It is also remarked there that we may apply any number k of such twists, provided that k ≡ 0 or 1 (mod 3) if n ≡ 3 (mod 12), and k ≡ 1 or 2 (mod 3) if n ≡ 7 (mod 12). 13 Embeddings and designs 275

Table 1.

Type 1 oriented triangles (j = 1, 3, 5,...,n− 2)

White Black (u0u0 u2 )(u0u1 ∞)(u0u2u1 ) j j+1 j+1 j j+1 j j j+1 (u1u1 u0 )(u1u2 ∞)(u1u0u2 ) j j+1 j+1 j j+1 j j j+1 (u2u2 u1 )(u2u0 ∞)(u2u1u0 ) j j+1 j+1 j j+1 j j j+1 (u0u1u2)(u0 u1 u2 ) j j j j+1 j+1 j+1

Type 2 oriented triangles (j = 1, 3, 5,...,n− 2)

Black (u0 u0 ∞) j+1 j+2 (u1 u1 ∞) j+1 j+2 (u2 u2 ∞) j+1 j+2

(All subscripts are modulo n − 1.)

Table 2.

Oriented triangles

White Black (u0u0 u1 )(u0u2 ∞)(u0u1u2 ) j j+1 j+1 j j+1 j j j+1 (u1u1 u2 )(u1u0 ∞)(u1u2u0 ) j j+1 j+1 j j+1 j j j+1 (u2u2 u0 )(u2u1 ∞)(u2u0u1 ) j j+1 j+1 j j+1 j j j+1 (u0u2u1)(u0 u2 u1 ) j j j j+1 j+1 j+1

We now make two observations about the construction that enable us to extend it. The proof of the original construction given in [34] continues to hold good for the extended version with obvious minor modiﬁcations. Firstly, the toroidal embedding of K3,3,3 given in Fig. 2 may be replaced by one in which the cyclic order of the three superscripts is reversed. The reversed embedding of K3,3,3 is isomorphic to the original but is labelled differently (see Fig. 3). For each white triangular face (uvw) of S we may carry out the bridging 0 1 2 operation across S ,S and S using either the original K3,3,3 embedding or the reversed embedding. The choice of which of the two to use can be made independently for each white triangle. 276 M. J. Grannell and T. S. Griggs

u0 v1 w0 u0

w1 u1 v2 v2

u2 v0 w2 w2

u0 v1 w0 u0

Fig. 3. Reversed toroidal embedding of K3,3,3

Secondly, it is not necessary for S0,S1 and S2 to be three copies of the same surface S. All that the construction requires is that the three surfaces have the ‘same’ white triangular faces and the ‘same’ cycle of n − 1 points around the border, all with the ‘same’ orientations. To be more precise, by the term ‘same’ we mean that there are mappings from the vertices of each surface onto the vertices of each of the other surfaces that preserve the white triangular faces, the border and the orientation. The sceptical reader may feel dubious that we can satisfy this requirement without in fact having three identically labelled copies of a single surface S. However, we shall see that not only is it possible to arrange this by other means, but it can often be done in many ways. The main result of [17] is the following:

2 Theorem 3.1 For n ≡ 7 or 19 (mod 36), there are at least 2n /54−O(n) non- isomorphic face 2-colourable triangulations of the complete graph Kn, and hence biembeddings of STS(n)s, in an orientable surface.

The basic idea of the proof is to use the construction described above with three fixed copies of the same embedding of Kn and a fixed auxiliary surface S, but varying the toroidal bridges. Since there are (n − 1)(n − 3)/6 bridges and two choices for each bridge, we may construct 2(n−1)(n−3)/6 differently labelled embeddings of K3n−2. From this it is possible to prove that, if S is suitably chosen, these embeddings are actually non-isomorphic. Thus the number of non- n2/6−O(n) isomorphic embeddings of K3n−2 is at least 2 , and replacing 3n − 2by n gives the result cited above. We now make an observation about the black triangles of the embeddings generated as described in Theorem 3.1. In any two such embeddings, the black 13 Embeddings and designs 277 triangles are identical and have the same orientations. To see this, note that the black triangles come from three sources. Those lying on the surfaces S0,S1 and S2 are unaltered during the construction and therefore are common to both embeddings. Those lying on the K3,3,3 bridges are the same whether or not the bridges are reversed (see Figs. 2 and3). Those lying on the surface S are common to both embeddings. It follows that the 2(n−1)(n−3)/6 non-isomorphic embeddings of K3n−2 all contain identical black triangles with the same orientations. In particular, the STS(3n − 2) defined by the black triangles is identical for each of these non-isomorphic embeddings. Wenext show how, by reapplying the construction, we can improve the exponent in Theorem 3.1 for some residue classes. In each of the embeddings of K3n−2, reverse the colours.Also, from each, delete the point ∞ together with all its incident edges and triangular faces. This produces a plentiful supply of non-isomorphic surfaces Si on which to base a reapplication of the construction in order to produce i embeddings of K9n−8. All of these surfaces S have the ‘same’ white triangles and the ‘same’ Hamiltonian cycle of points forming the border, all with the ‘same’ orientation. We can select three different surfaces from this collection to form N S0,S1 and S2 (in some order) in ways, where N = 2(n−1)(n−3)/6. We again 3 use a suitable fixed auxiliary surface S. The K3,3,3 bridges may be selected in (3n−3)(3n−5)/6 2 different ways. Any two of the resulting embeddings of K9n−8 0 1 2 (obtained by varying the surfaces S ,S and S , and the K3,3,3 bridges, but with a fixed S) are non-isomorphic. These results lead to the next theorem.

2 Theorem 3.2 For n ≡ 19 or 55 (mod 108), there are at least 22n /81−O(n) non- isomorphic face 2-colourable triangulations of the complete graph Kn, and hence biembeddings of STS(n)s, in an orientable surface. It is also shown in [17] that each of these embeddings has only the trivial automorphism. Similar results may be obtained in the non-orientable case. We form S0,S1 and S2 from three face 2-colourable embeddings (having the ‘same’ white triangles and the ‘same’ cycle of points around ∞)ofKn in a non-orientable surface. The white triangles are bridged using the toroidal embeddings given in Figs. 2 and3. The construction is completed, to form a face 2-coloured triangular embedding of

K3n−2 in a non-orientable surface, by forming a cap S with k twists, in the manner previously described. We must select k ≡ 1 or 2 (mod 3) if n ≡ 1 (mod 6), and k ≡ 0 or 1 (mod 3) if n ≡ 3 (mod 6). It was stated in [17] that there is a face 2-colourable triangular embedding of Kn in a non-orientable surface for each n ≡ 1 or 3 (mod 6) with n ≥ 9. At the time that paper was published, this does not seem to have been proved. However, the error was made good in [37], where the result was established. This enables us to state the following theorem from [17]. 278 M. J. Grannell and T. S. Griggs

2 Theorem 3.3 If n ≡ 1 or 7 (mod 18) and n ≥ 25, then there are at least 2n /54−O(n) non-isomorphic face 2-colourable triangulations of the complete graph Kn, and hence biembeddings of STS(n)s, in a non-orientable surface.

Once again we can make a colour reversal and then reapply the construction to form a face 2-colourable triangular embedding of K9n−8 in a non-orientable surface. Similar arguments to those given previously lead to the following theorem, again an amended version of a result of [17].

Theorem 3.4 If n ≡ 1 or 19 (mod 54) and n ≥ 73, then there are at least 2 22n /81−O(n) non-isomorphic face 2-colourable triangulations of the complete graph Kn, and hence biembeddings of STS(n)s, in a non-orientable surface.

Again, all of the embeddings of Theorem 3.4 are automorphism-free. To conclude this section we refer to some of the results given in [36]. One result generalizes the construction given above, and two other recursive constructions are presented. The generalization extends the earlier construction in suitable circumstances to produce biembeddings of two STS(m(n − 1) + 1)s from those of STS(n)s. Another construction parallels a well-known product construction for Steiner triple systems to produce a biembedding of STS(mn)s from one of STS(n)s, again subject to certain conditions. The third construction deals with biembeddings of transversal designs, deﬁned in an analogous manner to biembeddings of Steiner triple systems, so that suitable face 2-colourable triangulations of Kn,n,n and Km,m,m yield face 2-colourable triangulations of Kmn,mn,mn. All three constructions can be employed to give estimates, of a similar form to those above, of the numbers of non-isomorphic embeddings of Kn and of Kn,n,n for values of n in certain residue classes. It is worth noting that face 2-colourable triangulations of Kn,n,n, equivalent to biembeddings of transversal designsTD(3,n), are necessarily orientable as a consequence of the tripartition. For face 2-colourable triangulations of Kn, the estimates apply to both the orientable and the non-orientable cases.

4. Small systems

In this section we briefly summarize the current state of knowledge about biembeddings of STS(n)s for n = 3, 7, 9, 13 and 15. We re-emphasize that when referring to the number of biembeddings, we mean the number of non-isomorphic biembeddings of the specified type. When speaking of automorphisms, we include those that exchange the colour classes and, in the orientable case, those that reverse the orientation. The case n = 3 is trivial, since there is only one biembedding: it is orientable and has the automorphism group S3. The case n = 7 is less trivial, 13 Embeddings and designs 279 but well known. There is again only one biembedding: it is orientable and its automorphism group is the affine general linear group AGL(1, 7), which has order 42. In the realization shown in Fig. 1, this is z → az + b, a, b ∈ GF(7), a = 0. The automorphisms of even order exchange the colour classes but preserve the orientation. For n = 9 and n = 13, all biembeddings are necessarily non-orientable. The results for n = 9 and 13, and some of those for n = 15, were obtained by computer search. Two fixed systems are selected, one forming the black system and permutations of the other giving potential white systems. Since the black and the white systems cannot have a common triple, permutations giving rise to a common triple are discarded. This approach facilitated exhaustive searches in the cases n = 9 and n = 13, and partial searches in the case n = 15. The results cited below come from [28], [10], [11], [13] and [14], and most also appear in [8]. For n = 9, the biembedding is unique and has the automorphism group C3 × S3 of order 18. A realization is obtained by taking one system with block set {012, 345, 678, 036, 147, 258, 048, 156, 237, 057, 138, 246} and the other obtained from this by applying the permutation π = (01)(26)(47)(3)(5)(8). In this realization, the permutations π and(067)(184325)generate the automorphism group. The automorphisms of even order exchange the colour classes. There are two STS(13)s, one cyclic and the other not. We refer to these here as C and N, respectively. There are 615 biembeddings of C with C, of which 36 have an automorphism group of order 2 and four have an automorphism group of order 3; the rest have only the trivial automorphism. There are 8539 biembeddings of C with N, of which ten have an automorphism group of order 3 and the rest have only the trivial automorphism. Finally, there are 29 454 biembeddings of N with N, of which 238 have an automorphism group of order 2 and the rest have only the trivial automorphism. In each case, automorphisms of order 2 exchange the colour classes. We also note a paper of Ellingham and Stephens [25] in which they determine all non-orientable triangulations of K12 and K13, the latter including all face 2-colourable triangulations (that is, biembeddings of STS(13)s). The numbers are 182 200 for K12 and 243 088 286 for K13. For K12 there are, in addition, 59 orientable triangulations [3]. There are 80 non-isomorphic STS(15)s; a standard numbering and some of their structural features are given in [51]. A computer search has shown that each pair may be biembedded non-orientably [14], so there are at least 3240 non-orientable biembeddings (actually, far more). Almost all of those found have only the trivial automorphism group. Turning to orientable biembeddings of the STS(15)s, we firstly observe that there are precisely three systems with an automorphism of order 5. Each of these systems has an embedding with itself having an automorphism group of order 280 M. J. Grannell and T. S. Griggs

10. One of these was originally given by Ringel [56] and can also be obtained from Theorem 2.1. The other two may be obtained by Ringel’s method from current graphs [11]. In each case the automorphism of order 2 has a single fixed point, exchanges the colour classes, but preserves the orientation. In [15] a computer search for biembeddings of the 80 systems, each with itself, was based on examining all possible automorphisms of order 2 with a single fixed point and exchanging the colour classes. As a result, it was shown that 78 of the 80 systems have orientable biembeddings of this type. The exceptions are the systems numbered #2 and #79 in the standard listing. In the case of #2, it was further shown in [15] not to have an orientable biembedding with itself, and in [13] it was also shown that #1 cannot be orientably biembedded with #2. Again in [15], it was shown that if #79 biembeds with itself, then the biembedding can only have the trivial automorphism group. However, more recent work by the present authors and Martin Knor has disposed of this possibility. Hence we can state the following theorem. Theorem 4.1 Of the 80 non-isomorphic STS(15)s, 78 have a biembedding with themselves in an orientable surface. The two exceptions that have no such biembedding are #2 and #79 in the standard listing. An orientable biembedding of system #79 with system #77 having an automorphism of order 3 is also given in [15] and is the first known example of a biembedding of a pair of non-isomorphic STS(15)s though, as described in Section 5, there are already many known biembeddings of pairs of non-isomorphic STS(n)s for n = 19 and n = 31. Again, with Martin Knor, we have established a programme to find further such biembeddings. Of particular interest is whether there exists a biembedding of system #2 with some other system. In fact we have discovered such a biembedding and hence can state another theorem. Theorem 4.2 Each of the 80 non-isomorphic STS(15)s has a biembedding with some STS(15) in an orientable surface.

5. Cyclic embeddings

A triangular embedding of a graph in a surface may be described by means of a rotation scheme (see Chapter 1). Given a vertex x of the graph, the rotation about x comprises the cyclically ordered list of the vertices adjacent to x, taken in the order in which they appear around x in the embedding. The rotation scheme for the embedding is the set of all the vertices together with their rotations. In the orientable case, the rotations may be taken with a consistent orientation, that is, all clockwise or all anticlockwise. A rotation scheme is cyclic or of index 1 if we can denote the vertices by 0, 1,...,n− 1 in such a way that the rotation about x 13 Embeddings and designs 281 is obtained by adding x (mod n) to the rotation about 0. As we observed earlier, an orientable biembedding of two STS(n)s corresponds to a face 2-colourable triangular embedding of the complete graph Kn in an orientable surface, and it requires that n ≡ 3 or 7 (mod 12). In the case where n ≡ 3 (mod 12), a cyclic STS(n) contains a unique short orbit and consequently there can be no cyclic biembeddings. In [61], Youngs gives a cyclic orientable biembedding for all n ≡ 7 (mod 12), and it is this case that we consider here. We take as our starting point the result of [56] that every cyclic orientable embedding of K12s+7 can be derived from an appropriate current graph with 4s + 2 vertices. In our context, a current graph is a graph with directions (clockwise or anticlockwise) assigned at each vertex and whose edges are assigned both a direction (in the ordinary sense of the word) and a current, the current being a non-zero element of the group Z12s+7. An example for s = 2 is shown in Fig. 4.

2 13 10 6 7 1 14 15 3 4 12 9 11 5 8

Fig. 4. A current graph for s = 2

The rotation about 0 in the resulting embedding of K31 is obtained by traversing the graph, recording the (directed) currents encountered on each edge, and taking the clockwise or anticlockwise exit from each edge as indicated at that vertex. Thus we obtain the permutation

171529281022262082425527141623122118491911630217133.

The rotation about the vertex x is then obtained by adding x (mod 31) to each entry in this permutation. (A full explanation of current graphs is given in [38].) In the case where we are seeking an orientable biembedding of two STS(12s + 7)s the current graph must have the following properties.

(i) Each vertex has degree 3. (The graph is cubic.) (ii) At each vertex, the sum of the directed currents is 0 (mod 12s+7) (Kirchhoff’s current law). 282 M. J. Grannell and T. S. Griggs

(iii) Each of the elements 1, 2,...,6s + 3ofZ12s+7 appears exactly once as a current on one of the edges and each edge has exactly one of these currents. (iv) The directions (clockwise or anticlockwise) assigned to each vertex are such that a complete circuit is formed – that is, one in which every edge of the graph is traversed in each direction exactly once. (v) The graph is bipartite.

Properties (i) and (ii) ensure that the embedding is a triangulation, while properties (iii) and (iv) ensure that it is cyclic (see [56] and [38] for further details). Property (v) ensures that the embedding is face 2-colourable and therefore represents a biembedding of two STS(12s + 7)s. Consideration of the degree and the currents shows that these current graphs have 4s + 2 vertices. Furthermore, there can be no loops and (save for the exceptional case s = 0) no multiple edges. This last fact follows from consideration of the conﬁguration shown in Fig. 5.

x w

Fig. 5. A possible multiple edge.

If this forms part of a current graph, then w ≡ x, and so the whole current graph comprises two vertices with a triply repeated edge. There is a close connection between current graphs and solutions of Heffter’s ﬁrst difference problem (HDP). In 1897 Heffter [39] asked whether the integers 1, 2,...,3k can be partitioned into k triples a,b,c such that, for each triple, a + b ± c ≡ 0 (mod 6k + 1). Examination of the triples formed by the directed currents at each vertex in either of the two vertex sets of a bipartite current graph shows that they form a solution to HDP for k = 2s + 1. In view of the above observations, the problem of constructing cyclic orientable biembeddings of STS(12s + 7)s, s>0, may be reduced to three steps.

• Identify cubic bipartite simple connected graphs having 4s + 2 vertices. • Assign directions (clockwise or anticlockwise) at each of the vertices, which then give rise to a complete circuit. • Take two solutions of HDP and label the edges of the graph in such a way that the triples arising from each of the vertex-sets of the bipartition correspond to these two solutions. 13 Embeddings and designs 283

These three steps have a large measure of independence from one another. However, we cannot exclude the possibility that for a particular graph it may be impossible to assign vertex directions to give a complete circuit and, even if this is possible, it may not be possible to assign the HDP solutions to the edges. A test for the existence of a complete circuit in a graph G was given by Xuong [60]. It asserts the existence of such a circuit (equivalent to a one-face orientable embedding of G) if and only if G has a spanning tree whose co-tree has no component with an odd number of edges. Before proceeding further, we recall how Steiner triple systems arise from solutions to HDP. Given a difference triple a,b,c with a + b ± c ≡ 0 (mod 6k + 1), we can form a cyclic orbit by developing the starter {0,a,a+ b}orthe starter {0,b,a+b}. By taking all the difference triples from a solution of HDP and forming a cyclic orbit from each, a cyclic STS(6k + 1) is obtained. The converse is also true: given a cyclic STS(6k + 1), we can obtain a solution to HDP by taking from each orbit a block {0,α,β} and forming the difference triple ˆα,β− α,βˆ, where

x ≤ x ≤ k, xˆ = if 1 3 6k + 1 − x if 3k + 1 ≤ x ≤ 6k.

Each solution to HDP produces 2k different STS(6k + 1)s; however, there may be isomorphisms between these systems. In addition, for a given value of k, there will generally be many distinct solutions to HDP. For example, in [19] it is shown that for k = 3 there are four solutions to HDP, and these produce 4 × 23 distinct STS(19)s, which lie in four isomorphism classes. For n = 19, all the computations may be done by hand. The only cubic bipartite graph on six vertices is K3,3. Fixing the rotation about one vertex of K3,3, there are twelve ways of assigning vertex directions to produce a complete circuit. It is also easy to show that, from the four solutions to HDP, it is possible to obtain (up to isomorphism) only one pair of solutions with which to label the edges of K3,3 as described above. The resulting cyclic orientable biembeddings of STS(19)s are then found to lie in just eight isomorphism classes. The four cyclic STS(19)s are cyclically biembeddable, but none is cyclically biembeddable with itself. These embeddings were first given in [35] and further details of the argument sketched here appear in [12]. For n = 31, the computations require a computer. There are two cubic bipartite graphs on 10 vertices and they may be obtained from K5,5 by either removing a single 10-cycle, or a 6-cycle and a 4-cycle. Fixing the direction at one vertex gives a total of 160 sets of vertex directions in the former case and 128 sets of vertex directions in the latter case which result in complete circuits. Using the list of all solutions of HDP for k = 5 given in [19], we find 2408 isomorphism classes for 284 M. J. Grannell and T. S. Griggs cyclic orientable biembeddings of STS(31)s. There are 80 cyclic STS(31)s (see [22]), of which 76 are cyclically biembeddable. Of the 2408 isomorphism classes, 64 represent biembeddings of a system with itself and these involve 44 distinct systems. These were first given in [9] and further details of the argument again appear in [12]. For n = 43, there are 13 cubic bipartite graphs on 14 vertices to consider [55]. Of these, two have edge-connectivity 2, and so cannot have currents assigned along their edges that are different as required by property (iii) above. This is because the current in one of the two edges of the cutset would have to be equal (but opposite in direction) to that in the other. The 11 remaining graphs admit direction and current assignments. Further details are given in [12]. Before leaving this section we remark that [12] gives theoretical reasons, based on the above analysis, why certain pairs of cyclic STS(n)s cannot be cyclically biembedded together in an orientable surface. These are sufficient to give a complete explanation of cyclic biembeddability for n = 19 and 31, but not in general.

6. Concluding remarks

In this ﬁnal section, we note a variety of results related to our main theme of embedding Steiner triple systems. We also review some open problems. A particular interest of design theorists is the concept of a trade. Informally, this is a set T1 of blocks of an STS(n) for which it is possible to ﬁnd a disjoint set T2 of blocks (not lying in the system) which cover exactly the same pairs of points. The original STS(n) may then be transformed to a different, but possibly isomorphic, STS(n) by replacing T1 by T2. In [33] and [16], the authors investigate the analogous topological equivalent, where one set of triangles is replaced by a different set covering the same edges. In the recursive construction of Section 2, the replacement of a toroidal bridge (Fig. 2) by the reversed bridge (Fig. 3) is an example of such a topological trade. Indeed, Theorem 3.1 may be considered as a result about topological trades. Minimal trades between two orientable, between two non-orientable, and between orientable and non-orientable triangulations of Kn are determined in [33]. All topological trades involving m triangular faces for m ≤ 6 are described in [16]; there are none for m = 1, 2, 3 and 5, one for m = 4 and four for m = 6, each of which must lie in a limited number of geometrical patterns. For n lying in certain residue classes, Theorem 3.1 gives a lower bound of an2 the form 2 for the number of face 2-colourable triangulations of Kn in both orientable and non-orientable surfaces. Can this bound be extended to all possible residue classes? Is this the true order of growth? In a series of papers Korzhik and 13 Embeddings and designs 285

Voss ([45], [46], [47] and [43]) prove that for sufﬁciently large n, the number of βn orientable and non-orientable minimum genus embeddings of Kn is at least c2 , c> β> 1 K where 0 and 12 are constants. Minimum genus embeddings of n are triangulations when n ≡ 0 or 1 (mod 3) in the non-orientable case, and when n ≡ 0, 3, 4 or 7 (mod 12) in the orientable case. In the remaining cases, minimum genus embeddings of Kn are near-triangulations, having a small number of non- triangular faces. The basic technique employed in these papers relies on the use of appropriate current graphs. Although these results cover all residue classes, the 2 bound is a long way from 2an . In a more recent development, Korzhik and Kwak [44] combined the current graph approach with the cut-and-paste technique of Theorem 3.1 to prove that if 12s + 7 is prime and if n = (12s + 7√)(6s + 7), then n3/2( / +o( )) the number of non-orientable triangulations of Kn is at least 2 2 72 1 . Mao, Liu and Tian [50] give formulas for the numbers of embeddings of Kn in an orientable and in a non-orientable surface. However, no information is obtained about facial properties of the embeddings. We can obtain an upper estimate of the number of face 2-colourable triangulations by using the known upper bound for 2 the number of labelled Steiner triple systems of order n, namely (e−1/2n)n /6 [59]. Each labelled face 2-colourable orientable triangulation of Kn gives rise to a pair of labelled STS(n)s, ‘white’ and ‘black’. There are 2n(n−1)/6 possible choices for the orientations of the white triangles (that is, the blocks of the white system). Each such choice determines the orientation of the corresponding black triangles, so the number of labelled face 2-colourable orientable triangulations of Kn is at most

− / n2/ − / n2/ n(n− )/ n2/ (e 1 2n) 6(e 1 2n) 62 1 6

Consequently, the number of non-isomorphic face 2-colourable orientable n2/ triangulations of Kn is less than n 3. A similar argument may be applied in the non-orientable case. Unfortunately, there seems to be no simple way of using this type of argument to establish a lower bound, since an arbitrary pair of labelled STS(n)s is not, in general, biembeddable as the black and white systems of a face 2-colourable orientable triangulation of Kn, no matter what orientations are chosen for the blocks (for example, the systems may have a common triple). If the rate of growth of the number of non-isomorphic face 2-colourable triangulations an2 of Kn were of the order 2 , then this would imply that almost all STS(n)s are not biembeddable either orientably or non-orientably. Evidence culled from investigations of the 80 STS(15)s emboldens us to conjecture that, for n ≥ 9, each pair of STS(n)s is biembeddable in a non-orientable surface. However, it seems that the same may not be true in the orientable case, even with a ﬁnite number of exceptions; indeed, it is unclear whether, for each n ≡ 3 or 7 (mod 12), each STS(n) has an orientable biembedding with some other STS(n). 286 M. J. Grannell and T. S. Griggs

Lastly, one might reasonably consider variants of the above problems. For instance, embeddings of Kn could be sought in which the faces are all m-gons for some m>3. In the case m = n, the faces are Hamiltonian cycles, and Ellingham and Stephens [26] have shown that such Hamiltonian embeddings exist in a non-orientable surface for n ≥ 4,n = 5, and that for n odd the embeddings may be taken to be face 2-colourable. For graphs other than Kn a variety of results have been obtained. In particular, triangulations of Kn,n,n and related Hamiltonian embeddings of Kn,n are given in [1], [29], [30], [32] and [42], while triangulations related to biembeddings of symmetric conﬁgurations of triples (see [20] for terminology) appear in [52], [53] and [31].

Acknowledgements

Some parts of the text were originally published in [12], [17] and [35]. We thank our co-authors of these papers and the publishers for agreeing to their re-use here. We also thank the Leverhulme Trust for supporting our work with colleagues in Slovakia under the grant F/00269/E.

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MARK E. WATKINS

1. Introduction 2. Ends 3. Automorphisms 4. Connectivities 5. Growth 6. Inﬁnite planar graphs and maps References

Topological properties of infinite graphs may be global or local. The number of ends (equivalence classes of rays that cannot be separated by a finite subgraph) and whether a given end contains an infinite set of pairwise disjoint rays describe an infinite graph globally. Automorphisms are of interest in terms of both the cardinalities of their set of orbits as well as the cardinalities of the orbits themselves. The notion of connectivity is refined to consider whether the deletion of a subgraph leaves finite or infinite components. The rate of growth, whether polynomial or exponential, tells much about the graph’s global structure. Embedding of infinite graphs is of interest primarily in non-compact surfaces such as the plane, but even in the plane, issues arise concerning accumulation points. The interaction of these considerations is brought to bear on the structure of infinite planar graphs and maps.

1. Introduction

An infinite graph is locally finite if every vertex has finite degree. (Local finiteness does not imply that the set of degrees is bounded.) Generally, but not always, the graphs in this chapter are locally finite. A myopic graph theorist, standing at any vertex of an infinite but locally finite graph, views a landscape that resembles a finite

289 290 Mark E. Watkins graph of very large diameter. Thus, many of the ‘local’ properties of locally finite graphs coincide with analogous properties of finite graphs; the theories diverge only when one looks at infinite graphs ‘globally’. Some areas of infinite graph theory may reduce to trivialities in the case of finite graphs but require greater refinement in the case of infinite graphs. Non-trivial topological spaces can be constructed from various objects such as vertices, infinite paths, and sets of vertices or paths within an infinite graph itself, rather than, as in the case of finite graphs, mainly from the inherited topology of a surface into which a graph is embedded. This chapter will survey several areas of infinite graph theory that either directly involve topological spaces or are topological in flavour.

2. Ends

In order to describe the overall ‘shape’of an infinite graph (we’re speaking globally now), it is essential to speak of the ‘ends’ of the graph. This notion, previously used in describing infinite groups, was first applied to graphs by Halin [19]. In order to discuss ends, one must first consider the various kinds of paths that can occur in an infinite graph. A ray (also called a one-way infinite path)isa sequence of distinct vertices vi and edges ei, where ei = vi−1vi for all i ∈ N. A double ray (or two-way infinite path) is defined similarly, except that we have i ∈ Z. The word path is reserved henceforth for finite paths. We define a path, ray or double ray P in a graph G to be geodetic if for every v,w ∈ V(P), the distance equality dG(v, w) = dP (v, w) holds. Given a graph G, let R(G) denote the set of rays in G, and let R1,R2 ∈ R(G). We say that R1 and R2 are end-equivalent and write R1 ∼ R2 if there exists a ray R3 ∈ R(G) such that both V(R3 ∩ R1) and V(R3 ∩ R2) are infinite. It is not hard to see that ∼ is an equivalence relation on R(G). The equivalence classes with respect to ∼ are the ends of G. We denote by (G) the set of ends of G and let η(G) =|(G)|. The following result characterizes end-equivalent rays:

Theorem 2.1 Let R1,R2 ∈ R(G). The following statements are equivalent:

• R1 ∼ R2; • for any finite subset S ⊂ V (G), some component of G − S contains subrays of both R1 and R2; • G contains an infinite set of pairwise-disjoint paths joining R1 and R2. Thus, two rays belong to different ends exactly when they have subrays that can be separated by a finite set of vertices. When G is locally finite, η(G) has a convenient characterization; it is the supremum of the number of infinite components of G − S as S ranges over all 14 Infinite graphs and planar maps 291

finite subsets of V (G). In particular, G is 1-ended if and only if G − S has exactly one infinite component for any finite set S of vertices.

Examples Every regular tessellation of the plane has exactly one end, as does the Cartesian product of a ray with any connected locally finite graph. The Cartesian product of a double ray with any finite graph has exactly two ends. A double ray with a finite number of rays emanating from each of its vertices has countably many ends. The infinite 3-regular tree T3 has uncountably many ends. In fact, ℵ every locally finite graph has at most 2 0 ends. A complete bipartite graph with one finite partite set and one infinite partite set is ‘rayless’ and hence has no end, but a locally finite connected graph must contain a ray and hence have at least one end. This is a consequence of D. König’s Unendlichkeitslemma [36], which can be sharpened as in part (a) of the following theorem of Watkins [55].

Theorem 2.2 Let G be a locally ﬁnite connected graph. Then:

(a) for each vertex v ∈ V (G) and each end ω ∈ (G), there exists a geodetic ray emanating from v and belonging to ω; (b) for any two distinct ends ω1,ω2 ∈ (G), there exists a geodetic double ray that is the union of a ray from ω1 and a ray from ω2.

To appreciate the need for local finiteness in part (b), consider the union of two disjoint infinite complete graphs H1 and H2 together with a vertex v adjacent to every vertex in V(H1) ∪ V(H2). It has two ends and uncountably many rays but contains no geodetic ray. An end ω of G is dominated if for some v ∈ V (G) and every finite subset T ⊂ V (G)\{v}, G − T contains a ray emanating from v and belonging to ω. By the previous result, no end of a locally finite graph is dominated. When local finiteness is relaxed, we obtain the following more general statement of Polat and Watkins [43].

Theorem 2.3 For each vertex v and each non-dominated end ω of an inﬁnite connected graph, there exists a geodetic ray emanating from v and belonging to ω.

Halin [20] proved that each end ω of a graph G contains a set of pairwise-disjoint rays of maximum cardinality. In particular, if an end ω contains, for each n ∈ N,an n-set of pairwise-disjoint rays, then ω contains an inﬁnite set of pairwise-disjoint rays. The cardinality of a largest set of pairwise-disjoint rays in ω is denoted by µ(ω). (In [22], Halin proved an analogous result for double rays.) An end ω is thin if µ(ω) is ﬁnite; otherwise ω is thick. 292 Mark E. Watkins

The straightness of a ray or double ray R is deﬁned as d(v,w) σ(R) = lim inf , dR(v,w)→∞ dR(v, w) where v,w ∈ V(R). Thus 0 ≤ σ(R) ≤ 1. If R is geodetic, then σ(R) = 1, although not conversely (see Theorem 3.5(c) below). However, we say that R is quasi-geodetic if σ(R) > 0 or, equivalently, there exists some K>0 such that dR(v, w) ≤ Kd(v, w) for all v,w ∈ V(R)(see [32], where such rays or double rays are called ‘metric’). A ﬁner decomposition of R(G) than end-equivalence is due to Jung [30]. For rays R and S, let us write R ∼b S if there exist sequences of distinct vertices v1,v2,...∈ V(R)and w1,w2, ...∈ V(S)such that the three sets of numbers

{dR(vi,vi+1): i ∈ N}, {dS(wi,wi+1): i ∈ N}, {d(vi,wi): i ∈ N} are all bounded above. The relation ∼b is an equivalence relation, and the resulting equivalence classes of rays are called b-fibres. Between ∼b and ∼ in terms of refinement falls another equivalence relation ∼d called d-equivalence, due to Jung and Niemeyer (see [32], [40]). Here we have R ∼d S if for some M ∈ N, and for all v ∈ V(R)there exists some w ∈ V(S) such that d(v,w) ≤ M and for all w ∈ V(S)there exists some v ∈ V(R)such that d(v,w) ≤ M. The resulting equivalence classes of rays are called d-fibres. Jung and Niemeyer have proved the following: Theorem 2.4 Let G be a locally finite graph. Then: (a) in each b-fibre, either all rays are quasi-geodetic or no ray is quasi-geodetic; (b) every end of G contains at least one b-fibre consisting of quasi-geodetic rays; (c) quasi-geodetic rays are b-equivalent if and only if they are d-equivalent; (d) each end of G either is itself a d-fibre or is the union of uncountably many (disjoint)d-fibres. In the light of parts (a)–(c), we may speak of a geodetic fibre as a b-fibre − or, equivalently, a d-fibre − that contains a geodetic ray, and hence contains only quasi-geodetic rays. Thus, by Theorem 2.3, every end contains a geodetic fibre. There exists a topology that is definable in a natural way on the set (G) of a connected graph G. Let v ∈ V (G), and for each n ∈ N, let Hn denote the subgraph of G induced by the set of vertices at a distance greater than n from v. The neighbourhood system of an end ω ∈ (G) is the collection of sets Nn(ω) ⊂ (G) such that, for each n ∈ N, each end in Nn(ω) contains a ray belonging to the same component of Hn as a ray in ω. This neighbourhood system (as noted, for example, in [38]) induces on (G) a compact metrizable topological space called the end-space of G. 14 Infinite graphs and planar maps 293

In a paper of 1931, Freudenthal [12] defined the ‘ends’ of a topological space, from which Diestel and Kuhn [10] obtained a different approach to a topology of ends. Let the infinite graph G be regarded as a simplicial 1-complex, where vw typically each edge is regarded as a homeomorph of a closed interval in R. The basic open neighbourhood of each vertex v is a union {(v, w): vw ∈ E(G)} of half-open intervals. {X }∞ {Y }∞ In this topology of a simplicial complex, let i i=1 and i i=1 be decreasing nested sequences of non-empty open sets, each with a compact boundary, such that ∞ ∞ Xi = Yi = ∅. i=1 i=1 The topological ends of G (as opposed to the ‘graph-theoretical’ ends) are the equivalence classes of such sequences with respect to the equivalence relation that, for each i ∈ N, there exist j, k ∈ N such that Xi ⊇ Yj and Yi ⊇ Xk.In this context, Diestel and Kuhn demonstrated for any infinite graph G a natural bijection from the set of topological ends of G to the set of non-dominated ends in (G). Thus, for locally finite graphs, these two notions of ends are equivalent.

3. Automorphisms

A graph G is almost-transitive if Aut(G) acts on V (G) with finitely many orbits, in which case, if G is locally finite, Aut(G) also acts with only finitely many orbits on E(G). Even locally finite (and hence countable) graphs may have an uncountable automorphism group, as shown by Halin [23].

Theorem 3.1 Let G be a locally finite connected graph. Then Aut(G) is uncountable if and only if, for each finite subgraph H of G, there exists a non-identity automorphism that fixes every vertex of H .

In that same paper, Halin distinguished between ‘type 1 automorphisms’, those that fix (set-wise) some non-empty finite set of vertices, and ‘type 2 automorphisms’, that fix no such set. In subsequent work, the latter were renamed translations, which is a better description from an intuitive standpoint. The sets of vertices forming the orbits of a translation are of considerable interest. It is an easy exercise to show that if ϕ is an automorphism of a locally finite graph, then either all orbits of ϕ are infinite (when ϕ is a translation) or all orbits are finite (when ϕ is not a translation). It is also clear that automorphisms act as well-defined permutations on the sets of ends and the sets of fibres. Their action on the set (G) induces a group of homeomorphisms from the end-space of a graph G to itself. 294 Mark E. Watkins

Example The 3-regular tree T3 is clearly countable, but Aut(T3) is uncountable by Theorem 3.1. One can easily construct automorphisms of T3 that fix some vertex but have arbitrarily long (finite) orbits and hence have infinite order. An automorphism ϕ ∈ Aut(G) is bounded if there exists a constant Cϕ such that d(v,ϕ(v)) ≤ Cϕ for all v ∈ V (G). It is straightforward to show that the set B(G) of bounded automorphisms of G is a normal subgroup of Aut(G). Examples Consider the familiar square tessellation of the Euclidean plane whose vertices are ordered pairs (i, j) ∈ Z × Z. An automorphism τ is a translation if and only if it is of the form τ(i,j) = (i + k,j + ) for some k, ∈ Z. In this case, τ is bounded, with |k|+|| as a bound. Moreover, the translations are the only bounded non-identity automorphisms of this graph. On the other hand, consider the regular tessellation M = M(4, 5) of the hyperbolic plane, where all vertices have degree 4 and the boundaries of all faces are 5-gons. Let D be a double ray in M such that no face is incident with more than one edge of D. Then D is geodetic, and there exists a translation τ whose powers τ n for n ∈ Z \{0} are the only automorphisms that both advance D along itself and fix the two components of M − D. However, τ is not bounded. Each vertex v at distance d from D is mapped to a vertex τ(v)whose distance from v is given by an exponential function of d. In this case, the subgroup B(M) is trivial. Both of these examples are locally finite and vertex-transitive and show that, when η(G) = 1, translations sometimes do and sometimes do not belong to B(G). If G is almost-transitive and η(G) = 2, then B(G) must contain translations (see Theorem 4.7 below). However, in locally finite, infinitely-ended graphs, the group of bounded automorphisms contains no translation, as seen in the following result of Godsil et al. [13]. Theorem 3.2 Let G be a connected vertex-transitive locally finite graph with more than two ends. Then the orbits of B(G) are finite, all elements of B(G) have finite order, and all finitely generated subgroups of B(G) are finite. Every translation τ of a locally finite connected graph fixes at least one end. There exists a ray R that passes successively through the sequence of vertices v,τ(v),τ2(v), . . . and has the property that τ(R) ⊂ R. The end containing R is called the direction of τ and is denoted by D(τ). Thus τ distinguishes up to two (not necessarily distinct) ends, D(τ) and D(τ −1), and there exists a τ-invariant double ray that is the union of two rays, one from each of these distinguished ends. Moreover, these are the only ends of G fixed by τ. Clearly D(τ) = D(τ n), for all n ∈ N. A major structural theorem of Halin [23] contains the following additional information. Theorem 3.3 Let τ be a translation of a locally finite connected graph. Then the following hold: 14 Infinite graphs and planar maps 295

• µ(D(τ)) = µ(D(τ −1)); • D(τ) = D(τ −1) if and only if D(τ) is thick; • G contains a set of µ(D(τ)) pairwise-disjoint double rays, each of which is fixed by τ; • if D(τ) is thin, then there exists a set T of µ(D) vertices such that no component of G − T contains both a ray from D(τ) and a ray from D(τ −1). Without the constraint of local connectivity, Polat and Watkins [43] proved the following two results: Theorem 3.4 Let τ be a translation of a connected graph G whose direction is a thin end. Then for some r such that 1 ≤ r ≤ µ(D(τ)), there exists a τ r -invariant geodetic double ray which is the union of two rays, one from D(τ) and the other from D(τ −1). Theorem 3.5 Let the translation τ ∈ Aut(G) fix some quasi-geodetic double ray D0 in G. Then the following hold: (a) every double ray in G fixed by a non-zero power of τ is also quasi-geodetic; (b) if σ(D0)<1, then some power of τ fixes a quasi-geodetic double ray D1 with σ(D1)>σ(D0); (c) if σ(D0) = 1, then D0 is geodetic. In the light of Theorem 3.5(b), it is natural to ask: is it always the case that sup{σ(D): τ n(D) = D for some n ∈ N}=1? An affirmative answer was given by Jung and Niemeyer [32] in the case of locally finite graphs. To conclude this section, we state a seminal result relating ends and automorphisms of locally finite graphs. As a corollary to Theorem 3.3, Halin [23] proved the following powerful restriction to the possibilities for the number of ends of a graph: Corollary Let G be a connected locally finite graph such that Aut(G) contains a translation. Then η(G) = 1, 2 or ∞. In particular, η(G) = 2 if and only if G contains no infinite set of pairwise disjoint rays. This powerful corollary leads one to seek a sufficient condition for a locally finite connected graph to admit a translation. Such a condition requires the notion of a ‘fragment’ (see Jung [29]), which will be defined in the next section.

4. Connectivities

For an infinite graph G, we define its finite-connectivity by

κf (G) = inf {|S|: S ⊂ V (G) and G − S has a ﬁnite component}. 296 Mark E. Watkins

If G has a vertex of finite degree, then clearly κf (G) < ∞.Aκf -separator of G is a set S ⊂ V (G) such that |S|=κf (G) and G − S has a finite component. In locally finite graphs, finite-connectivity κf has many of the properties of connectivity in finite graphs. The global counterpart of κf is infinite-connectivity, defined as follows:

κ∞(G) = inf {|S|: S ⊂ V (G); G − S has at least two inﬁnite components}.

This makes sense except when G has no infinite component. If there is no finite (possibly empty) subset of V (G) whose deletion leaves at least two infinite components, then we write κ∞(G) =∞.Aκ∞-separator of G is a set S ⊂ V (G) such that |S|=κ∞(G) and G − S has at least two infinite components. (These terms were introduced in [33].) The connectivity of G is defined by

κ(G) = min{κf (G), κ∞(G)} and agrees with the definition of connectivity for finite graphs. Example Let D be a double ray, let n ≥ 3, and consider the Cartesian product D Cn. Then κf (D Cn) = deg(D Cn) = 4, while κ∞(D Cn) = n, showing that either of these connectivities may be the larger one. Here the κf - separators are the neighbourhoods of the vertices, and the κ∞-separators are copies of Cn. On the other hand, for the lexicographic product D[Cn], the κ∞-separators are the same as for the Cartesian product, but κf (D[Cn]) = 2n and each κf -separator is the union of two non-adjacent copies of Cn. While graphs that are not locally finite may have uncountably many κ∞-separators, every vertex of finite degree belongs to only finitely many κ∞-separators. The κ∞-separators of a locally finite graph are distributed in a more orderly fashion (see Jung and Watkins [35]). Theorem 4.1 Let G be a connected, locally finite graph, and let S and T be distinct κ∞-separators of G. Then: • T meets no finite component of G − S; • if G − S has more than two infinite components, then T meets exactly one of them. For any set F of vertices of G, we define the sets

∂F ={x ∈ V \F : x is adjacent to some vertex in F } and F = V \(F ∪ ∂F).

Thus κ∞(G) = min{|∂(F)|: F and F are infinite} whenever this value is finite; otherwise κ∞ =∞. If both F and F are infinite and |∂F|=κ∞(G), then F is 14 Infinite graphs and planar maps 297 called a fragment of G (see [28]); in this case, ∂F = ∂F . It is straightforward to prove that the intersection of two fragments F1 and F2 is a fragment, provided that both F1 ∩ F2 and F1 ∩ F2 are infinite. The following result is due to Babai and Watkins [1]: Theorem 4.2 Let G be a connected infinite locally finite graph. If Aut(G) contains a torsion subgroup that acts almost-transitively, then κ∞(G) =∞. From this theorem, Jung [29] deduced that if a graph has finite connectivity and admits a vertex-transitive group of automorphisms, then it is locally finite. He went on to provide the sufficient condition alluded to at the end of the previous section.

Theorem 4.3 If κ∞(G) < ∞ and L is a vertex-transitive subgroup of Aut(G), then for each fragment F of G, there exists some λ ∈ L such that λ(F ∪∂F) ⊆ F . In the language of this theorem, we see that among the ends containing a ray in F is the direction D(λ) of λ, which must be a thin end. One can intuitively imagine successive positive powers of λ as shoving D(λ) ever further into itself. Thus, λ must have an infinite orbit. In fact, λ would do as the translation mentioned in Theorem 3.6. Putting these two results together we obtain something extremely useful. Corollary If G is locally finite, connected and vertex-transitive, then η(G) = 1, 2 or ∞. ℵ One may replace ∞ by 2 0 in this corollary and may replace ‘vertex-transitive’ by ‘almost-transitive’in both this corollary and Theorem 4.3. The corollary allows most problems concerning almost-transitive, locally finite graphs to decompose naturally into three qualitatively different cases, each of which in turn determines a great deal about the graph’s behaviour.

If 0 <κf (G) < ∞, we define an atom to be a finite component of G − T with the least number of vertices as T ranges over all the κf -separators of G. We define α(G) =|V (A)|, where A is any atom. Clearly the atoms of G are non-trivial (that is, α(G) > 1) if and only if κf (G) is less than the minimum degree. This notion was first used in conjunction with finite graphs (see [54] and [37]), where the following were shown: • Distinct atoms are disjoint. • If G is edge-transitive, then κ(G) equals the minimum degree. • If G is vertex-transitive and connected, then: G is a union of disjoint copies of the same atom together with edges joining the copies; each atom is itself a vertex-transitive subgraph; 298 Mark E. Watkins

each minimum separating set is also a union of atoms. • {κ(G)/ (G) G }= 2 inf deg : is vertex-transitive and connected 3 and this bound is never attained.

These results all carry over from ﬁnite graphs to locally ﬁnite graphs when κ is replaced by κf . Suppressing the argument G, let us write κf = qα+r for integers q ≥ 1 and 0 ≤ r<α. Then, for any vertex v belonging to an atom, we can generalize the last bulleted item as follows, even without any assumption about transitivity: deg(v) ≤[(q + 1)/q]κf − 1 − r/q. (1)

For any atom A and any κf -orκ∞-separator T , we have either A ⊆ T or A∩T = ∅.IfG is a union of atoms (as, for example, when G is vertex-transitive), then equality holds in (1) if and only if G is a lexicographic product of the form ˆ G[Kα], in which case q ≥ 2 and r = 0, as shown in [33].

Examples Let D be a double ray, and let G be the lexicographic product D[Kn]. The copies of Kn are the atoms, and they are induced by the κ∞-separators of G. The κf -separators are the unions of any two non-adjacent κ∞-separators. Thus µ(G) = κ∞(G) = n, and κf (G)/deg(G) = 2n/(3n − 1), illustrating how the 2 inﬁmum of 3 is approached but never attained. On the other hand, the strip D Kn has only trivial atoms – that is, α(D Kn) = 1. Then κ(D Kn) = κ∞(D Kn) = n holds, because κf (D Kn) = deg(D Kn) = n + 1. 2 Babai and Watkins [1] showed that a larger inﬁmum than 3 occurs when certain conditions hold for the group Aut(G).

Theorem 4.4 If G is a connected, r-regular graph (r < ∞), and if a torsion (G) V (G) κ(G) ≥ 3 (r + ) subgroup of Aut acts transitively on , then 4 1 and the edge-connectivity of G equals r.

The two product graphs in the preceding examples are ‘strips’. A connected graph G is a strip if it contains a connected subgraph H and admits an automorphism ϕ such that 0 < |∂H| < ∞, ϕ(H ∪ ∂H) ⊆ H , and H − ϕ(H) is ﬁnite. If we knew that |∂H|=κ∞(G), then we could conclude that H is a fragment and that ϕ is a translation. The latter is in fact true, and H is (modulo a ﬁnite subgraph) a fragment. The following characterization of a strip is due to Jung and Watkins [34].

Theorem 4.5 An infinite connected graph G is a strip if and only if it is locally finite and Aut(G) contains an automorphism with finitely many orbits.

Imrich and Seifter [26] gave another characterization of strips. 14 Inﬁnite graphs and planar maps 299

Theorem 4.6 A graph G is a strip if and only if η(G) = 2 and Aut(G) contains a translation. A characterization of strips in terms of ‘growth’ will be given in the next section. Planar strips may be regarded as the covering graphs of almost-transitive graphs embeddable on the torus, whose vertices are precisely the orbits of the automorphism described in Theorem 4.5. The following result of Jung [31] shows that strips can be ubiquitous as induced subgraphs of multi-ended graphs. Theorem 4.7 Let G be locally finite and suppose that some subgroup L ≤ Aut(G) fixes some two ends and contains a translation. Then G contains an induced L-invariant strip S such that every component of G − S has a finite boundary. The automorphism group of a strip is often not very complicated, as indicated by the next result of Jung and Watkins [34]. Theorem 4.8 Let G be an an infinite connected graph, and suppose that Aut(G) contains an Abelian subgroup A acting transitively on V (G). Then either ∼ κ∞(G) =∞,orG is a strip. In the latter case, A = Z × F for some finite group F . Edge-transitive strips G have a very specific structure in which the subgroup B(G) of bounded automorphisms plays a distinctive role. The next theorem from Watkins [56] describes their structure.

Theorem 4.9 Let G be an edge-transitive strip. Let T be a κ∞-separator of G, let v ∈ T , and let N(v) be the set of neighbours of v. Let Autv(G) denote the stabilizer of the vertex v. Then: • all vertices of G have even degree; • if G is not vertex-transitive or if G is arc-transitive, then the index |Aut(G): B(G)|=2 and Autv(G) acts transitively on N(v); • if G is vertex-transitive but not arc-transitive, then either Aut(G) = B(G) and Autv(G) has two orbits in N(v),or|Aut(G): B(G)|=2 and Autv(G) has two or four orbits in N(v). If an edge-transitive strip is planar, then it has a very special structure. Corollary If G is a planar edge-transitive strip, then κ(G) ≤ 4. If κ(G) > 2, then κf (G) = deg(G) = 4, G is arc-transitive, and all faces in any planar embedding are 4-gons. The graphs described in this corollary can be visualized as taking the square tessellation of the Euclidean plane and ‘rolling up’the plane along an axis at 45◦ to the major axes. It follows that all (ﬁnite) vertex- and edge-transitive toroidal graphs 300 Mark E. Watkins of connectivity greater than 2 are 4-regular and dually 4-regular. In fact, they are self-dual. This special class of graphs will be revisited below in Theorem 6.7.

5. Growth

In this section, without further mention, G denotes exclusively an infinite locally finite graph. Let v ∈ V (G). For n ∈ N ∪{0}, we define

Sn(v) ={w ∈ V (G): d(v,w) = n};

Bn(v) ={w ∈ V (G): d(v,w) ≤ n};

f (n, v) =|Bn(v)|.

Thus f(0,v)= 1 and f(1,v)= deg(v)+1. We call Sn(v) the n-sphere and Bn(v) the n-ball about v. The function f (n, v) is the growth function of G with respect to the vertex v. Clearly the growth functions with respect to vertices in the same orbit of Aut(G) are identical, and if G is vertex-transitive, then we may suppress the vertex v altogether. We are generally interested in the asymptotic behaviour of f (n, v) as n →∞. When G is connected, this is always computable and is independent of the reference vertex v; that is to say, there always exists a function g such that f (n, v) = O(g(n)) for all v ∈ V (G). Hence, when G is connected, we suppress the vertex v and write brieﬂy f (n). On the other hand, limn→∞ |Sn(v)| need not exist.

Example Suppose that for some vertex v ∈ V (G) and for all n ∈ N, S2n(v) induces the complete graph Kn while S2n−1(v) consists of a single vertex of degree 2n−1. Clearly f (n, v) = O(n2), while |Sn(v)| is asymptotic to no familiar function. n If limn→∞ f (n)/c = 0 for some constant c>1, then G has subexponential growth; otherwise, G has exponential growth. As a particular case of subexponential growth, G has polynomial growth and its growth degree is δ if δ = inf {d: f (n) ≤ cnd for all n ∈ N} for some constant c. A graph with linear growth has growth degree 1; if the growth degree is 2, then its growth is quadratic. That δ is always an integer follows from a result of Gromov [16]. Graphs with subexponential growth that exceeds any polynomial growth have intermediate growth. These descriptive words for rates of growth of graphs derive from similar usage for rates of growth of groups (see [16]). If G is connected and r-valent, then for any vertex v,

n−1 |Sn(v)|≤r(r − 1) , for all n ∈ N, (2) 14 Inﬁnite graphs and planar maps 301 from which it follows that

n n (r − 1) − 1 n f (n) = |Sj (j, v)|≤1 + r = O(r ). (3) r − 2 j=0

Examples For the square tessellation of the Euclidean plane, we have |Sn|= 4n and f (n) = 1 + 2(n2 + n) for all n ≥ 1; thus the square tessellation (or 2-dimensional lattice) has quadratic growth. One can show that, in general, the d-dimensional lattice has polynomial growth with growth degree d. For the infinite r-regular tree Tr , equality holds in (2) and hence also in (3). Hence Tr has exponential growth. Cayley graphs of groups of intermediate growth are graphs of intermediate growth. Such groups are not finitely representable, and their Cayley graphs are definitely not planar. Putting these examples together, we see that infinite planar graphs may have linear, quadratic or exponential growth. These are, in fact, the only possible rates of growth possible for almost-transitive planar graphs. The following are some typical results concerning graphs of linear or quadratic growth. (For a survey of the properties of graphs of polynomial growth, see Imrich and Seifter [27].) The first is another characterization of strips due to Imrich and Seifter [26]; compare Theorem 4.7.

Theorem 5.1 A connected locally ﬁnite vertex-transitive graph has linear growth if and only if it is a strip.

Theorem 5.2 The automorphism group of any almost-transitive graph with quadratic growth contains a subgroup isomorphic to Z2 that acts almost- transitively.

From Theorem 5.2, Seifter and Troﬁmov [47] concluded that there are only countably many vertex-transitive graphs (up to isomorphism) with at most quadratic growth. For polynomial growth degree d ≥ 3, sufﬁcient conditions are not so forthcoming for Zd to act almost-transitively (see [45]). However, Godsil et al. [13] proved the following result about all graphs with polynomial growth.

Theorem 5.3 Let G be a connected locally ﬁnite vertex-transitive graph. If G has polynomial growth, then the action of Aut(G) is imprimitive.

The second part of the following result of Seifter [46] gives another necessary and sufﬁcient condition for graphs with polynomial growth to have an uncountable automorphism group that acts vertex-transitively (compare Theorem 3.1). 302 Mark E. Watkins

Theorem 5.4 Let G be a connected locally ﬁnite vertex-transitive graph with polynomial growth. Then:

• no ﬁnitely generated group of intermediate growth can act transitively on V (G); • Aut(G) is uncountable if and only if Aut(G) contains a ﬁnitely generated subgroup with exponential growth that acts transitively on V (G).

A classic theorem of Tutte ([53], p. 59) states that every finite vertex-transitive, edge-transitive graph of odd degree is also arc-transitive. Tutte inquired whether the condition of odd degree could be relaxed. A negative answer to Tutte’s query was provided by Bouwer [7], who constructed an infinite family of (finite) counter- examples. Tutte’s theorem is extendable to graphs of subexponential growth, but counter-examples exist if the degree is allowed to be even or if the growth is allowed to be exponential (see Thomassen and Watkins [52]). A d-fibre F is translatable if there exist a translation τ ∈ Aut(G) and a ray R ∈ F such that τ(R) ⊂ R. A graph G is periodic if some finitely generated abelian subgroup A ≤ Aut(G) acts almost-transitively on V (G). It is not difficult to show that the automorphism group of a periodic graph contains an isomorphic copy of some Zk that acts almost-transitively on V (G). Furthermore, if k is the largest such integer, then G has polynomial growth of degree k, and k is called the rank of G. The following two theorems of Niemeyer and Watkins [42] bring together several of the concepts presented in this chapter.

Theorem 5.5 Every periodic graph G has either ﬁnitely many (but at least 2) or uncountably many geodetic d-ﬁbres. Furthermore,

• if G has finitely many geodetic d-fibres, then all the d-fibres are translatable; in particular, G has exactly two geodetic d-fibres if and only if G is a strip – that is, its rank is 1. • if G has uncountably many geodetic d-fibres, then all but finitely many of them contain uncountably many geodetic rays (sharing no common subray) and countably many of these d-fibres are translatable.

Theorem 5.6 Every planar 3-connected periodic graph has either exactly two or uncountably many geodetic d-ﬁbres.

The second of these two theorems rightly suggests that many of the results stated thus far in this chapter have a much stronger conclusion when the hypothesis of planarity is added. This leads us to the ﬁnal section. 14 Inﬁnite graphs and planar maps 303

6. Inﬁnite planar graphs and maps

Every connected non-planar graph of finite genus must contain a finite subgraph whose deletion leaves a planar graph. This was shown by Imrich and Seifter [25], who proved further the following result. Theorem 6.1 All 3-connected locally finite non-planar infinite graphs of finite genus have finite automorphism groups. It follows directly that the only locally finite almost-transitive graphs of finite genus are planar. The rules for embedding an infinite graph G in the plane to form a planar map necessarily go beyond the rules for embedding a finite graph in the plane, because locally finite infinite maps still must deal with issues concerning ‘infinite regions’. When a finite graph G is embedded in a surface S, it is usual to define a ‘region’ to be a component of S\G. One needs only to assume that G is 2-connected to assure that the boundary of every region is a cycle. This definition is inadequate for infinite maps. For example, consider the square tessellation of the plane embedded in an open unit square Q, which is in turn a subset in the plane. The complement of Q in the plane would be a region according to the definition for finite maps, but what would be its boundary? Or, consider a planar embedding of the infinite ladder D K2 (where D is a double ray), where the ladder is ‘folded’ at a copy of K2 and ‘half’ of the ladder is reflected into the interior of a 4-cycle. Where and how many are the limit points of the reflected half-ladder? Since any embedded cycle in G together with its interior is a compact subset of the plane, and since such a subset may very well contain the embedding of a connected infinite subgraph of G, we must take care where the accumulation points of the embedded graph occur. We distinguish two kinds of accumulation points. A vertex-accumulation point is simply an accumulation point of the embedded vertex-set of the graph. If e1,e2,...is a sequence of distinct edges of G and, for each i ∈ N, Pi is a point in the embedding of ei, then any accumulation point of the set {Pi: i ∈ N} is an edge-accumulation point of G. While accumulation points and infinite regions may be permissible and often inevitable, one feels that they are somehow undesirable aesthetically and therefore that one should attempt to minimize them. To this end, we impose the following three additional requirements proposed by Richter [44] (listed in [5]) for an embedding of an infinite graph in the plane: (a) No vertex- or edge-accumulation point is also a point of the embedded graph. (b) The set of vertex- and edge-accumulation points is a nowhere dense subset of the plane. This implies that, if e1,e2, ...is a sequence of distinct edges of G and if some sequence P1,P2,..., where Pi is a point in the embedding of ei, 304 Mark E. Watkins

converges to a point P0, then every sequence {Qi: i ∈ N} such that Qi is a point on ei also converges to P0. Thus, every edge-accumulation point is also a vertex-accumulation point, and so we may speak simply of an accumulation point of G. (c) Every region is incident with at most one accumulation point of the embedding. (Thus a component of the complement of G cannot contain more than one region.)

We further require that the boundary of every connected component of the complement of G be a connected subgraph of G. Thus, by requirement (c), the boundary of every region must be either a closed walk or a two-way infinite walk. If G is 2-connected, then the boundary of every face is either a cycle or a double ray. This requirement, in effect, defines what is acceptable as a ‘region’. By way of notation, if G is a planar graph, then its embedding in the plane is called a map M = M(G). Its vertex-set V(M) and edge-set E(M) are the sets of embedded vertices and edges, respectively, of G. The set of faces arising from this embedding will be denoted by F(M). The cardinality of a set of edges incident with a face f is the dual degree or co-degree of f , where cut-edges are double-counted. Graphs that are not locally finite often cannot be embedded in the plane to conform to the above embedding rules. Moreover, several of the results stated below fail for graphs that are not locally finite. Therefore, as in Section 5, all infinite graphs encountered are assumed to be locally finite. A different approach to embedding infinite graphs in the plane is taken by Bonnington, Richter and Watkins [6]. Issues involving accumulation points are obviated by using only the discrete metric topology of the graph and the rotational orders of edges about the vertices rather than the continuous topology of the surface; in fact, ‘faces’ are not mentioned at all. In this context we define a facial walk (...,vi−1,ei,vi,ei+1,...) to be a walk determined by the following two conditions:

• in the clockwise rotation about each vertex vi, the edge ei+1 immediately follows the edge ei; • the walk is maximal with respect to the property that no ordered pair (vi,ei+1) appears more than once.

Thus, a facial walk all of whose vertices are distinct is either a double ray or a cycle. A well-known theorem of H. Whitney [57] (see Chapter 1) states that if a 3-connected finite graph G is planar, then the cyclic order of the edges incident with each vertex about that vertex is the same in every map M(G) and that this property determines which cycles are the boundaries of regions. These conclusions 14 Infinite graphs and planar maps 305 were extended to 3-connected infinite graphs independently by Imrich [24] and Thomassen [50]. A map automorphism of the map M = M(G) not only is an automorphism of its underlying graph G, but also induces a permutation of the set F(M) of faces of M.IfG is 3-connected, then every automorphism of G is extendable to an automorphism of M. We say that M is face-transitive if Aut(M) acts transitively on F(M). Dirac and Schuster [11] attribute to Paul Erdos˝ a proof of the following statement: if every finite subgraph of G is planar, then G is planar. Thus the absence of a homeomorph of either of the Kuratowski graphs K5 and K3,3 as a subgraph of G is necessary and sufficient for the planarity of G. Dirac and Schuster pointed out that if ‘planar’ were to be defined as being embeddable in the plane without accumulation points, then this criterion would fail. In this vein, Halin [21] proved that an infinite graph is embeddable in the plane without accumulation points if and only if it contains no homeomorph of any of eight configurations (two of which are the Kuratowski graphs). If η(G) = 1, then no cycle separates two infinite subgraphs, and so G is embeddable in such a way that for any cycle C of G, the unique infinite component of M(G) − C is in the exterior of C. The following result is from Bonnington, Imrich and Watkins [4]:

Theorem 6.2 If G is a planar 2-connected 1-ended and almost-transitive graph, then, in every planar embedding of G, the boundary of every region is a cycle.

In [6] it is shown that for an inﬁnite 2-connected planar map, the number of inﬁnite faces cannot exceed the number of ends.

Examples The 2-way infinite ‘ladder’ D K2, where D is a double ray, has two ends and admits an obvious planar embedding with two infinite faces and no accumulation point. While all planar embeddings of the 2-ended graph D Ck (k ≥ 3) admit a planar embedding with no infinite faces, there is always at least one accumulation point. In this case, consecutive copies of Ck are embedded concentrically and converge to that point. The map M(D Ck) is vertex-transitive and self-dual but not edge-transitive. The Cartesian product T3 K2 (where T3 is the 3-regular tree) is a vertex- transitive planar graph with κ = κ∞ = 2. However, no vertex-transitive subgroup ofAut(T3 K2) is extendable to a group of map-automorphisms of M(T3 K2), as shown by Bonnington and Watkins [5]. By Theorem 6.2, the square lattice in its usual depiction has neither accumulation points nor infinite faces. To construct an embedding with a single accumulation point and one of its 4-gons f as the exterior face, first project the 306 Mark E. Watkins plane (containing the square lattice with no accumulation point) onto the sphere punctured at the North Pole so that the South Pole lies in the interior of f . Then rotate the sphere so that the two poles swap positions and project the sphere back onto the plane. Thus the image of f becomes the exterior face and the image of the South Pole is the unique accumulation point. In [55], Watkins proved that if a geodetic double ray D is deleted from a vertex-transitive graph G, then all remaining components are infinite. How many such infinite components can there be if G is planar? In particular, when does D split M(G) into exactly two parts? What happens if D is only quasi-geodetic? How much symmetry is required of G? From Bonnington, Imrich and Watkins [4] comes the following theorem, and no geodetic assumption for D is needed. (Note that almost-transitive graphs have bounded co-degrees.)

Theorem 6.3 Let G be planar and 1-ended. Let D be a double ray in G. Then:

(a) if G is 2-connected and almost-transitive, then G − D has at most two inﬁnite components; (b) if G is 3-connected and has bounded co-degrees and if D is quasi-geodetic, then G − D has exactly two inﬁnite components.

The two inﬁnite components mentioned in part (b) must lie on opposite ‘sides’ of D, in a way that is intuitively clear but can be made explicit in terms of the cyclic order of edges about any vertex on D (see [4], p. 69). This notion of ‘plane separation’can be used to characterize planarity for a large class of 1-ended graphs. For this purpose, we say that G is almost-4-connected if κf (G) ≥ 3 and, for every κf -separator T , every ﬁnite component of G − T is a single vertex. Thus every 1-ended almost-4-connected graph is 3-connected. The following limited characterization of planarity comes from Bonnington et al. [4]; its proof is adapted from a proof by Thomassen [51].

Theorem 6.4 Let G be a 1-ended almost-transitive almost-4-connected graph. A necessary and sufﬁcient condition for G to be planar is that, for every quasi- geodetic double ray D, G − D has exactly two inﬁnite components.

Counter-examples to the sufficiency of this condition exist, on the one hand, if ‘almost-4-connected is weakened to ‘3-connected’ and, on the other hand, if ‘quasi-geodetic’is sharpened to ’geodetic’. One is led to inquire whether this same condition is equivalent to planarity under the assumption only that G is 1-ended and vertex-transitive. Bonnington, Imrich and Seifter [3] gave an affirmative answer when G has polynomial growth, but the question remains open otherwise. A geodetic edge of a graph is an edge that lies on some geodetic double ray. Bonnington, Imrich and Seifter [3] conjectured that every edge of a 1-ended vertex-transitive planar graph is a geodetic edge and showed the conjecture to 14 Infinite graphs and planar maps 307 hold for bipartite graphs. They further credited Carsten Thomassen with a proof for 3-regular graphs. When κf = 3, vertex-transitivity may not be weakened to almost-transitivity, as shown by the following example. Starting with the regular triangular tessellation of the Euclidean plane, insert a new vertex in the interior of each region, joining it by an edge to each of the three vertices on the boundary of that region. These new edges are not geodetic. But there is more to this story. Let Ga,b denote the class of all 1-ended 3-connected planar maps, all of whose degrees and co-degrees are finite and at least a and at least b, respectively. Let Ga+,b denote the subclass of Ga,b in which no two vertices of degree a are adjacent. Similarly, let Ga,b+ denote the subclass of Ga,b in which no two faces of co-degree b share an edge. It was shown by Niemeyer and Watkins [41] that all edges of maps in G4,4 are geodetic, and Bruce [8] has done the same for the classes G3,6, G6,3 and G5, 3+ . For all of these classes of maps, no transitivity condition was required. Nonetheless, the conjecture remains open when both vertices of degree 3 and faces of co-degree 3 are present.An important tool in obtaining these results is the notion of a ‘Bilinski diagram’, after S. Bilinski [2]. It is a manner of labelling the sets of vertices and faces of a 1-ended map with no infinite faces. Bilinski diagrams have been used extensively by Grünbaum, particularly in his book with Shephard [17]. Let G be a 1-ended planar 3-connected graph, and let v ∈ V (G). A map B = M(G) is labelled as a Bilinski diagram of G (and G as the underlying graph of B) with respect to the following notation:

• U0 ={v}; v is the root of B; • F1 is the set of faces incident with x; • for m ≥ 1, Um is the set of those vertices not in Um−1 that are incident with a face in Fm; • for m ≥ 1, Fm+1 is the set of faces not in Fm that are incident with a vertex in Um. A Bilinski diagram B so labelled is concentric if for all m ≥ 1, the induced subgraph Um is a cycle. A graph G is universally concentric if every Bilinski diagram of G (that is, regardless of the choice of the root) is concentric. Grünbaum and Shephard [17] showed that every regular tessellation of the Euclidean and hyperbolic planes admits some concentric Bilinski diagram. Since the graphs that underlie regular tessellations are vertex-transitive, they are universally concentric. However, all graphs in Ga,b, for (a, b) = (3, 6), (6, 3), (4, 4), (3+, 5) or (5, 3+), are also universally concentric (see [41], [8]), and some of them are very ‘irregular’. If G has more than one end and if B is a Bilinski diagram of G, then for some sufﬁciently large m there would exist a ﬁnite separating set that contains Um. Hence for some n>m, the submap induced by Un would not be connected. Thus no multi-ended graph admits a concentric Bilinski diagram. 308 Mark E. Watkins

G It is interesting and perhaps counter-intuitive to note that, if is almost- O(| n U |) = f (n) f (n) transitive, then m=0 m , where is the growth function defined in Section 5 (see Moran [39]). Generalizing Coxeter [9], we define a Petrie walk in a map (on any orientable surface) to be a maximal walk in which every two consecutive edges are incident with a common face but no three consecutive edges have this property. We stipulate further that no two edges appear consecutively in the same order more than once. Thus a Petrie walk is either closed or two-way infinite. (Petrie walks have been also called ‘left-right paths’ by Shank [48].) Each edge of a map belongs to either one or two Petrie walks. A nice property of Petrie walks is that if W is a Petrie walk in a map M, then the dual edges of W in the dual map M∗ form a Petrie walk in M∗. Coxeter [9] observed that in the nine finite 3-connected edge-transitive planar maps, the Petrie walks are cycles of even length. In infinite 2-connected edge- transitive planar maps, every Petrie walk is either a cycle of even length or a double ray, and every edge lies on exactly two distinct Petrie walks (see Graver and Watkins [14]). All Petrie walks in planar embeddings of graphs in G3,6 ∪ G4,4 are geodetic double rays (see Bruce [8] and Niemeyer and Watkins [41]). Let M be a finite or infinite 2-connected planar map. Following Grünbaum and Shephard [18], we assign to each edge e of M its edge-symbol r1,r2; s1,s2, where r1 and r2 are the degrees of the two vertices incident with e, and s1 and s2 are the co-degrees of the two faces incident with e. If all edges of M have the same edge-symbol (aside from interchanging r1 with r2 or s1 with s2), then that is the edge-symbol of M, and M is edge-homogeneous. An edge-homogenous planar map with symbol r1,r2; s1,s2 has exponential growth if the quantity ζ = 1 + 1 + 1 + 1 r r s s 1 2 1 2 is less than 1, has linear or quadratic growth if ζ = 1, and is finite in the nine realizable cases where ζ>1. Clearly, if M is edge-transitive, then it is edge-homogeneous. Grünbaum and Shephard [18] considered the converse implication:

Theorem 6.5 There exists an edge-homogeneous 3-connected planar map with r ,r ; s ,s r ,r ,s ,s edge-symbol 1 2 1 2 if and only if all of 1 2 1 2 are integers at least 3 and exactly one of the following holds:

• r ,r ,s ,s all of 1 2 1 2 are even; • s = s r r 1 2 is even, and at least one of 1 and 2 is odd; • r = r s s 1 2 is even, and at least one of 1 and 2 is odd; • r = r s = s 1 2 , 1 2 , and all are odd. 14 Inﬁnite graphs and planar maps 309

If such a map is 1-ended, then it is edge-transitive and is determined, up r = r to isomorphism, by its edge-symbol. If, moreover, 1 2 , then the map is s = s arc-transitive, and if 1 2 , then it is face-transitive.

Let M be a 3-connected edge-transitive planar map. Since there are exactly two Petrie walks through each edge, Aut(M) induces either one or two orbits of Petrie walks, called Petrie-orbits. Following [14], we say that M is of cycle type if both Petrie walks through each edge are even cycles; M is of ray type if both walks are double rays; M is of mixed type when one Petrie-orbit consists of even cycles and the other consists of double rays. There are strong correlations between ‘type’, the number of ends and, to a lesser extent, the edge-symbol. For example, Graver and Watkins [14] proved the following result.

Theorem 6.6 An inﬁnite 3-connected map M is of cycle type if and only if M or its planar dual has edge-symbol 4, 4; 3,h for some h ≥ 6. Furthermore, every graph of cycle type is 1-ended.

If h<6, then h = 3or4,andM is finite. Edge-transitive 1-ended maps whose edge-symbol is not of the form 4, 4; 3,h or 3,h; 4, 4 for h ≥ 6 are all of ray type. By Theorem 4.6 and the Corollary to Theorem 4.9, edge-transitive 2-ended planar maps are strips. We describe their construction explicitly. For each k ≥ 2, let Mk denote the quotient graph of the planar tessellation with edge-symbol 4, 4; 4, 4 (coordinatized as the integer lattice Z × Z) modulo the identification (i, j) ≡ (i + k,j + k) for all (i, j) ∈ Z × Z. Thus Mk is a 2-ended edge-transitive planar map with κ∞(Mk) = k. It has the same edge-symbol 4, 4; 4, 4 and is arc-transitive. Intuitively, imagine that an infinitely long circular cylinder has been placed on the Cartesian plane at a 45◦ angle to the coordinate axes. The edge-transitive strip Mk is then obtained by rolling up the plane around the cylinder so that, for each pair (i, j), all the vertices (i + ck, j + ck) as √c runs through Z are identified. The circumference of the cylinder would thus be 2 · κ∞(Mk) units. The cylinder is homeomorphic to the punctured plane, and the one point to be deleted (that is, the puncture) is the one accumulation point of the planar map. (Compare the remark at the conclusion of Section 4.) These graphs are of mixed type; one Petrie walk through any given edge of Mk consists of a cycle of length 2κ∞(Mk) zigzagging around the cylinder, and the other is a double ray running longitudinally on the cylinder. To summarize (see Watkins [56]):

Theorem 6.7 For each integer k ≥ 2, Mk is the unique 2-ended edge-transitive planar map with all degrees at least 3 and κ∞(Mk) = k. 310 Mark E. Watkins

One of the achievements in [14] is a method, called ‘interleaving’, for constructing edge-transitive infinitely-ended planar maps. Some of these are of ray type and some are of mixed type. The existence of edge-transitive infinitely-ended planar maps that are not constructible by interleaving remains an open question. An anomaly of infinitely-ended edge-transitive planar maps is that, unlike the 1-ended case described in Theorem 6.5, the edge-symbol r1,r2; s1,s2 does not uniquely characterize the map. Let z denote the length of a shortest cycle in M whose deletion leaves at least two (in fact, exactly two) infinite components. (It is not trivial that such cycles exist.) Let us call r1,r2; s1,s2; z the extended edge- symbol of M. It can be shown in some cases that, if r1 = r2 and s1 = s2, then r1,r2; s1,s2; z is the extended edge-symbol of exactly two (non-isomorphic) edge-transitive, infinitely-ended planar maps. In conclusion, we mention some areas of ongoing research on locally finite planar maps. One of these concerns the rate of growth of those 1-ended planar maps upon which some kind of homogeneity has been imposed. In the case of edge-transitivity, we refer the reader to Graves, Pisanski and Watkins [15]. Other kinds of homogeneity also present interesting unsolved problems. Let σ =s1,s2,...,sk be a cyclic sequence of integers greater than 2. If every vertex v of a 1-ended planar map M has the property that σ describes the co-degrees of the faces incident with v as one proceeds clockwise or anticlockwise about the vertex v, then M is vertex-homogeneous. Questions arise for given σ concerning realizability and uniqueness. If σ is realizable as a map M,isM vertex-transitive? If M is vertex-transitive, is its underlying graph a Cayley graph? Is there a subgroup of Aut(M) that both preserves orientation and acts transitively on V(M)? These questions have been answered when k = 3, 4 and 5 (see Šiagiová and Watkins [49]), but many open questions remain, because no theorem analogous to Theorem 6.5 for edge-homogeneous maps is known for vertex-homogeneous maps. The dual of vertex-homogeneity is face-homogeneity. In this context, Moran [37] has studied the question of ‘stability’ – that is, when a well-defined growth function of the associated Bilinski diagrams exists. However, much is unknown concerning the growth of maps with various other kinds of homogeneity.

References

1. L. Babai and M. E. Watkins, Connectivity of infinite graphs having a transitive torsion group action, Arch. Math. 34 (1980), 90–96. 2. S. Bilinski, Homogene Netze der Ebene, Bull. Internat. Acad. Yougoslave Cl. Sci. Math. Phys. Tech. 2 (1949), 63–111. 3. C. P. Bonnington, W. Imrich and N. Seifter, Geodesics in transitive graphs, J. Combin. Theory (B) 67 (1996), 12–33. 4. C. P. Bonnington, W. Imrich and M. E. Watkins, Separating double rays in locally finite planar graphs, Discrete Math. 145 (1995), 61–72. 14 Infinite graphs and planar maps 311

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31. H. A. Jung, On finite fixed sets in infinite graphs, Discrete Math. 131 (1994), 115–125. 32. H. A. Jung and P. Niemeyer, Decomposing ends of locally finite graphs, Math. Nachr. 174 (1995), 185–202. 33. H. A. Jung and M. E. Watkins, On the connectivities of finite and infinite graphs, Monatsh. Math. 83 (1977), 121–131. 34. H. A. Jung and M. E. Watkins, Fragments and automorphisms of infinite graphs, European J. Combin. 5 (1984), 149–162. 35. H. A. Jung and M. E. Watkins, Finite separating sets in locally finite graphs, J. Combin. Theory (B) 59 (1993), 15–25. 36. D. König, Theorie der endlichen und unendlichen Graphen, Akad. Verlagsgesellschaft, 1936. 37. W. Mader, Über den Zusammenhang symmetrischer Graphen, Arch. Math. 21 (1970), 331–336. 38. B. Mohar, Some relations between analytic and geometric properties of infinite graphs, Discrete Math. 95 (1991), 193–219. 39. J. F. Moran, The growth rate and balance of homogeneous tilings in the hyperbolic plane, Discrete Math. 173 (1997), 151–186. 40. P. Niemeyer, Verfeinerung des Endenbegriffes bei unendlichen Graphen, Diplom thesis, TU-Berlin, 1992. 41. P. Niemeyer and M. E. Watkins, Geodetic rays and fibers in one-ended planar graphs, J. Combin. Theory (B) 69 (1997), 142–163. 42. P. Niemeyer and M. E. Watkins, Geodetic rays and fibers in periodic graphs, J. Graph Theory 34 (2000), 67–88. 43. N. Polat and M. E. Watkins, On translations of double rays in graphs, Period. Math. Hungar. 30 (1995), 145–154. 44. R. B. Richter, personal communication. 45. N. Seifter, Groups acting on graphs with polynomial growth, Discrete. Math. 89 (1991), 269–280. 46. N. Seifter, Properties of graphs with polynomial growth, J. Combin. Theory (B) 52 (1991), 222–235. 47. N. Seifter and V. I. Trofimov, Automorphism groups of graphs with quadratic growth, J. Combin. Theory (B) 71 (1997), 205–210. 48. H. Shank, The theory of left-right paths, Combinatorial Mathematics, III (eds. A. P. Street and W. D. Wallis), Lecture Notes in Math. 452, Springer-Verlag (1975), 42–54. 49. J. Šiagiová and M. E. Watkins, Covalence sequences of planar vertex-homogeneous maps, Discrete Math. 307 (2007), 599–614. 50. C. Thomassen, Duality of infinite graphs, J. Combin. Theory (B) 33 (1982), 137–160. 51. C. Thomassen, The Hadwiger number of infinite vertex-transitive graphs, Combinatorica 12 (1992), 481–491. 52. C. Thomassen and M. E. Watkins, Infinite vertex-transitive, edge-transitive, non 1- transitive graphs, Proc. Amer. Math. Soc. 105 (1989), 258–261. 53. W. T. Tutte, Connectivity in Graphs, Univ. of Toronto Press, 1966. 54. M. E. Watkins, Connectivity of transitive graphs, J. Combin. Theory 8 (1970), 23–29. 55. M. E. Watkins, Infinite paths that contain only shortest paths, J. Combin. Theory (B) 41 (1986), 341–355. 56. M. E. Watkins, Edge-transitive strips, Discrete. Math. 95 (1991), 359–372. 57. H. Whitney, Non-separable and planar graphs, Trans. Amer. Math. Soc. 34 (1932), 339–362. 15 Open problems

DAN ARCHDEACON

1. Introduction 2. Drawings and crossings 3. Genus and obstructions 4. Cycles and factors 5. Colourings and ﬂows 6. Local planarity 7. Thickness, book embeddings and covering graphs 8. Geometrical topics 9. Algorithms 10. Inﬁnite graphs References

We present a variety of open problems in topological graph theory, ranging through classical questions on genus, map colourings, crossing numbers, and geometrical representations. We also consider algorithmic questions and inﬁnite graphs.

1. Introduction

We all have our problems. In this chapter I present some of my favourite ones in topological graph theory. Some of these come from a problem list [3] I have been maintaining at http://www.cems.uvm.edu/∼archdeac/problems, others are new; all, I think, are interesting. A. N. Whitehead said ‘The “silly” question is the ﬁrst intimation of some totally new development’; I hope that some of these questions are silly. In Section 2 we give some problems on drawing graphs in the plane, including the number of crossings. Section 3 contains some classical problems on genus and

313 314 Dan Archdeacon on obstructions to embeddings. In Section 4 we turn our attention to finding cycles, factors and trees in embedded graphs. A look at colouring and flow problems is in Section 5, while Section 6 examines problems involving local planarity, also known as representativity. Section 7 then covers the thickness of graphs, book embeddings and planar covers. Other geometrical topics are studied in Section 8, and algorithmic problems in Section 9. We finish in Section 10 with a look at infinite graphs. We give a small sample of open and (we hope) interesting problems in topological graph theory. You are invited to explore the literature, the web and your imagination for others.

2. Drawings and crossings

A drawing of a graph in the plane has its vertices represented by distinct points and an edge vw represented by a simple curve joining the points representing v and w. No point representing a vertex can be an interior point of a curve representing an edge, and any two edges share a ﬁnite number of points. Moreover, when two distinct edges share an interior point, their geometrical curves cross transversally. A drawing is good if

• no two edges share more than a single interior point; • two adjacent edges do not share an interior point; • no three edges cross at the same point.

The usual goal is to draw a graph in the plane with the minimum number of crossing points. The ﬁrst two conditions ensure that there are no unnecessary crossings, while the third ensures that counting the intersection points is the same as counting pairwise crossings. The crossing number cr(G) is the minimum number of edge crossings over all good planar drawings of G.

Complete graphs Our ﬁrst conjecture is about the crossing number of the complete graph. (K ) = 1 1 n 1 (n − ) 1 (n − ) 1 (n − ). Conjecture 2.1 cr n 4 2 2 1 2 2 2 3 The conjectured crossing number is known to be an upper bound. We ﬁrst describe a drawing with this number of crossings when n = 2m. Place m vertices regularly spaced along both the top and bottom of a tin can (a cylinder homeomorphic to a sphere). Two vertices along the top rim are connected along the lid with a straight-line segment; similarly for two vertices along the bottom rim. 15 Open problems 315

A vertex on the top is connected to one on the bottom with a lateral line segment of minimum possible positive winding number around the cylinder. For n = 2m − 1 we delete one vertex from the drawing described above to achieve the conjectured minimum. The conjecture is known to be true for n ≤ 12 (see [32], [60]). If the conjecture is true for an odd value of n, then it is also true for n + 1; this follows from an argument counting, with multiplicities, the number of crossings in a drawing of all Kns contained in an optimal drawing of Kn+1. It would be interesting to show that the conjectured upper bound is (K )/n4 = 1 asymptotically correct: that is, cr n 64 . The best-known upper bound is due to E. de Klerk, D. Pasechnik, R. B. Richter and G. Salazar [46] who show that this limit is at least 0.83 of its conjectured value using arguments similar to that described above. Suppose that we are given a drawing of the complete graph Kn in the plane. This drawing determines a local rotation at each vertex: the clockwise cyclic permutation of the edges incident with each vertex from 1 to n. This collection of local rotations is called a rotation. For example, there are sixteen possible rotations on K4, and eight of these correspond to drawings in the plane, two of them without crossings.

Problem 2.2 Determine those rotations on Kn that can arise from a planar drawing.

One approach is to use the rotation to count the number of crossings in a drawing. A solution to the above problem would reduce ﬁnding the minimum crossing number of the complete graph to a purely combinatorial problem.

Complete bipartite graphs The following is the conjectured crossing number of the complete bipartite graph.

(K ) =1 m 1 (m − ) 1 n 1 (n − ) Conjecture 2.3 cr m,n 2 2 1 2 2 1 . This problem is historically known as Turán’s brickyard problem, where the vertices in the two parts represent kilns and warehouses, and the edges are tracks joining all of the kilns to each of the warehouses. The conjectured bound can be 1 n realized by the following drawing. Place 2 vertices along both the positive and x 1 n 1 n n 1 m negative -axis (or 2 and 2 , respectively, if is odd), and 2 vertices along the positive and negative y-axis (again splitting nearly equally if m is odd). Now connect each pair of vertices on different axes with straight line segments. The difﬁculty lies in showing that every other drawing has at least as many crossings as this. 316 Dan Archdeacon

The problem was thought to have been solved by Zarankiewicz, but his proof was derailed by Guy [32]. If n ≤ m, then the bound is tight for n ≤ 6 [45] and for n = 7 and m ≤ 10 [92].

Geometrical (linear) crossing number A variation on the crossing number requires all edges to be straight line segments. The minimum number of edge crossings over all drawings of this type is called the geometrical crossing number cr(G) of G. Drawings of this type are also called linear or rectilinear [58] and the minimum number of crossings is then called the linear crossing number or rectilinear crossing number. Clearly cr(G) ≥ cr(G). Strict inequality can hold: for example, Richard Guy has shown that 19 = cr(K8)>cr(K8) = 18, and similarly, 62 = cr(K10)>cr(K10) = 60. (The ﬁrst equality is shown in [13].) Problem 2.4 Find the geometrical crossing number of the complete graph.

Problem 2.5 Find limn→∞ cr(Kn)/n4 and limn→∞ cr(Kn)/cr(Kn). The ﬁrst limit is greater than 0.0158345, which is very close to the upper bound (see [1]). Lovász, Vesztergombi, Wagner and Welzl [49] showed that the second limit is strictly greater than 1, so the geometrical crossing number exceeds the crossing number of Kn for large enough n. It is believed that cr(Kn)>cr(Kn) for all n ≥ 10. In contrast to the complete graph, we have the following conjecture. Conjecture 2.6 The geometrical crossing number of the complete bipartite graph is equal to its crossing number.

Maximum crossing number Most of the time, one tries to minimize the number of crossings in a good drawing. What happens if we examine the opposite extreme: ﬁnding a good drawing of a graph that maximizes the number of crossings? A thrackle is a drawing of a graph in which any two non-adjacent edges cross exactly once. Conjecture 2.7 If a graph can be thrackled, then the number of edges does not exceed the number of vertices.

The converse is not true, as is shown by a 4-cycle. Woodall [91] has shown that every cycle of length at least 5 can be thrackled, as can every tree. In fact, this Thrackle conjecture reduces to showing that the one-vertex union of two cycles of even lengths cannot be thrackled. Cairns and Nikolayevsky [15] showed that 3 (n − ) if a graph can be thrackled, then the number of edges is at most 2 1 . John 15 Open problems 317

Conway offers a prize of $1000, or 1000 units in the currency of your choice, for a solution to Conjecture 2.7. See [3] for more history of this problem. The maximum crossing number crM (G) of a graph G is the maximum number of pairwise edge-crossings, taken over all good drawings of G.

Conjecture 2.8 If H is a subgraph of G, then crM (H ) ≤ crM (G).

One approach might be to take a drawing of H achieving the maximum crossing number and try to extend it to a drawing of G; the resulting drawing, if it existed, would provide the desired lower bound on the maximum crossing number of G. However, such an extension does not always exist (see [64]). Nonetheless, I believe that this conjecture is true. For some other interesting problems on crossings, see [59].

3. Genus and obstructions

The orientable genus of a graph G is the minimum number of handles of an orientable surface S such that G embeds in S: that is, such that G can be drawn on S without crossings. The non-orientable genus is the minimum number of crosscaps of a non-orientable surface in which G embeds. Determining the orientable and non-orientable genus of the complete graph is commonly known as the map colour problem [66]. This motivated the examination of the genus of many common graphs: complete bipartite graphs, n-dimensional cubes, etc.

Finding triangulations Thomassen [82] showed that it is an NP-complete problem to determine if a given graph G triangulates a surface. However, by the map colour theorem [66], complete graphs triangulate an orientable surface provided that the numerical constraints from Euler’s formula are satisﬁed. Mohar and Thomassen [53] asked whether this extends to graphs that are ‘close enough’ to being complete.

Problem 3.1 Does there exist a number c, with 0

We ask the corresponding question for non-orientable surfaces. I believe that the answer to both questions is ‘yes’. More strongly, I think that for every ε>0, N c = 1 + ε n ≥ N there is an integer such that 2 sufﬁces for all . 318 Dan Archdeacon

Background on obstructions We next consider the class of graphs that embed on a fixed surface. For example, Kuratowski’s theorem [48] characterizes planar graphs: they are those that contain no topological copy of K5 or K3,3. We examine similar obstruction theorems. Any type of obstruction theorem concerns partial orders and hereditary properties. The first such order is the subgraph order. Planarity is hereditary under this order – that is, if H ≤ G and G is planar, then H is planar. The topological ordering allows the taking of subgraphs and the suppression of vertices of degree 2 (removing a vertex v of degree 2 incident with edges uv, vw and adding an edge uw). The minor ordering allows these operations and also the contraction of edges; note that contracting an edge incident with a vertex v of degree 2 is the same as supressing v. The Y- ordering additionally allows the deletion of a vertex v of degree 3 followed by the addition of three edges joining pairwise the three vertices adjacent to v. Finally, the H- ordering also allows the operation of subdividing an edge joining two vertices v,w of degree 3, followed by two Y- operations on v and w. Planarity is hereditary under each of these successively finer partial orders. An obstruction to a hereditary property P is a graph G without property P, but such that each H

The torus We now focus on the simplest non-spherical orientable surface: the torus.

Problem 3.2 Find the minor-minimal non-toroidal graphs.

There are several partial lists of obstructions compiled under different orders (see [10], [19], [43], [37], [57]). If a complete set of obstructions is known for any one of the above partial orders, then a straightforward computation gives the complete set of obstructions under the other partial orders. The ﬁner partial orders 15 Open problems 319 have fewer obstructions. Bodendiek and Wagner estimated that under the H - order there are about 30 obstructions. The best lists to date are due to Gagarin, Myrvold and Chambers [27] who have found 239 322 obstructions for the torus under the topological order, and 16 629 obstructions under the minor order. Perhaps a more tractable problem is the following, which would characterize cubic graphs that are embeddable on the torus.

Problem 3.3 Find the cubic topological-minimal non-toroidal graphs.

The best list known to date is due to Myrvold, who has found all 206 cubic obstructions with up to 24 vertices. There may not be any obstructions of larger order, as there are only two such obstructions with 22 vertices and two with 24 vertices. The set of minor-minimal non-toroidal graphs is known to be ﬁnite, by the graph minor theorem of Robertson and Seymour [68], [69], but their work does not give a good bound on the size. Mohar and Thomassen [53] asked for such a bound in the following special case.

Problem 3.4 Does every minimal forbidden subgraph for the torus have fewer than 100 edges?

Other surfaces The non-orientable genus γ˜ (G) of a graph G is the minimum number k such that G embeds in the surface formed by adding k crosscaps to the sphere. It is known that the deletion of an edge may lower the non-orientable genus by 2 – that is, there exists a graph G containing an edge e such that γ(G˜ − e) =˜γ (G) − 2. Bodendiek and Wagner [10] asked the following question.

Question 3.5 Does there exist a graph such that the deletion of every edge lowers the non-orientable genus by 2?

This question is equivalent to: ‘Does there exist a graph that is a minimal obstruction to embedding in two different non-orientable surfaces?’ One approach would be to examine graphs whose automorphism group acts transitively on the edges; this would reduce the difficulty in checking the genus of the edge-deleted subgraphs. The spindle is created by identifying two distinct points on a sphere to form a single pinch point. A neighbourhood of this pinch point is not homeomorphic to the plane: it has two sheets, so is a spindle not a surface. A pseudosurface is formed by the arbitrary identification of points on (not-necessarily-connected) surfaces, so that there are a finite number of pinch points each with a finite number of sheets. The property of embedding in a fixed pseudosurface is not hereditary 320 Dan Archdeacon under the minor order, but embedding in the spindle is hereditary. Hence we ask the following: Problem 3.6 Find the minor-minimal graphs that do not embed on the spindle.

The corresponding problem has been solved [6] for cubic graphs under the topological order.

4. Cycles and factors

There are a wide variety of problems about ﬁnding cycles and factors in embedded graphs. We begin with one of the most important areas.

Cycle double covers Let G be a 2-connected graph embedded in the plane. The boundary of each face is a simple cycle, and all such face-boundaries collectively contain each edge exactly twice. The following conjecture is due independently to Szekeres [75] and Seymour [72]. Conjecture 4.1 Every graph without cut-edges has a collection of cycles that together contain each edge exactly twice.

Solving the conjecture is almost easy. Form G from G by replacing each edge by two parallel edges. Then in G each vertex has even degree and so G has an edge-partition into cycles. However, some of these cycles may be of length 2 and hence do not correspond to cycles in G. A stumbling block to inductive proofs has been found in many different contexts – namely, suppose that each edge e is assigned a weight w(e) = 1or 2 so that the sum of the weights at each vertex is even. Can we ﬁnd a cycle cover so that each edge e is used w(e) times? The answer is no: a counter-example is formed from the Petersen graph by assigning weight 1 to two disjoint 5-cycles and weight 2 to the remaining 5 edges. The preceding problem is not entirely topological, although its motivation is. The following topological variation is called the circular embedding conjecture [33].

Conjecture 4.2 Every 2-connected graph has an embedding in which each face is bounded by a simple cycle.

The face-boundaries of such an embedding form a cycle double cover. Conjectures 4.1 and 4.2 are equivalent for cubic graphs, but Conjecture 4.2 is stronger in general. Jaeger [40] made the stronger conjecture: 15 Open problems 321

Conjecture 4.3 In some embedding of a 2-connected graph in an orientable surface, each face is bounded by a simple cycle.

An even stronger conjecture asserts that there is an embedding as in Conjecture 4.3, where the faces can be properly 5-coloured. The following Petersen colouring conjecture [41] implies Conjecture 4.2 and hence also Conjecture 4.1.

Conjecture 4.4 Let G be a bridgeless cubic graph and let P be the Petersen graph. Then there is a homomorphism from the line graph L(G) to L(P ).

Partial results are too numerous to mention here: see the survey article by Jaeger [40] and the more recent book by Zhang [95].

Matchings A perfect matching or 1-factor in a graph is a collection of edges that together are incident with each vertex exactly once. A 1-factorization is a partition of the edges into 1-factors. If the graph is regular of degree d, then a 1-factorization is equivalent to a proper d-edge-colouring. Not every cubic graph can be edge-partitioned into perfect matchings; for example, the Petersen graph cannot be. The following was conjectured by Berge and by Fulkerson [26]:

Conjecture 4.5 Every bridgeless cubic graph has a collection of six perfect matchings that together contain each edge exactly twice.

The Petersen graph is not a counter-example: the six 1-factors form such a double cover. Note that any 3-edge-colourable cubic graph trivially satisﬁes the conjecture. Seymour [73] generalized Conjecture 4.5 to r-regular graphs with a suitable generalization of cut-edge-free.

Hamiltonian cycles Tutte [86] proved that every 4-connected planar graph is Hamiltonian. We look for similar results on other surfaces. Thomas and Yu [76] proved that every 4- connected projective-planar graph is Hamiltonian, and also (see [77]) that every 5-connected toroidal graph is Hamiltonian. The following question is due to Grünbaum [31] and Nash-Williams [55].

Question 4.6 Is every 4-connected toroidal graph Hamiltonian?

Thomas,Yuand Zang [78] showed that every such graph contains a Hamiltonian path. For some other questions along these lines, see Section 6 of this chapter. We mention the following conjecture of Barnette. 322 Dan Archdeacon

Conjecture 4.7 Every cubic 3-connected bipartite planar graph is Hamiltonian.

Other spanning subgraphs AHamiltonian cycle is just one example of a spanning subgraph.Weask about other types. A tree without vertices of degree 2 is called homeomorphically irreducible. The next conjecture is due to Malkevitch [50].

Conjecture 4.8 Every 4-connected planar graph has a homeomorphically irreducible spanning tree.

It is known that any planar triangulation [2] or toroidal triangulation [17] has a homeomorphically irreducible spanning tree, but there are 3-connected 4-regular planar graphs without them.

Conjecture 4.9 Every graph that triangulates some surface has a homeomorphically irreducible spanning tree.

More broadly, the conjecture may hold for graphs with every edge in at least two triangles or, less broadly, it may hold for surface triangulations of large width (see Section 6 below for deﬁnitions). AHalin graph is a 3-connected planar graph with a distinguished face f incident only with vertices of degree 3, such that deleting the edges incident with f leaves a tree. The leaves of the tree are necessarily the vertices incident with f . Plummer [61] conjectured the following.

Conjecture 4.10 Every 4-connected plane triangulation has a spanning Halin subgraph.

The analogous conjecture for arbitrary 4-connected plane graphs is false.

5. Colourings and ﬂows

Much of modern graph theory was motivated by the four-colour problem; for a history of this problem and its solution we refer the reader to Wilson [90]. All existing solutions to this problem involve ﬁnding a large unavoidable set of reducible conﬁgurations, and then verifying the reducibility with a computer.

The four-colour theorem and its relatives We begin with the following.

Problem 5.1 Find a non-computer proof of the four-colour theorem. 15 Open problems 323

The following conjecture, probably due to S. Fisk, is a strong version of the four-colour theorem: Conjecture 5.2 The number of 4-colourings of a plane triangulation with no vertices of degree 3 goes to inﬁnity as the number of vertices goes to inﬁnity.

Thomassen asserted a similar conjecture for graphs without a K4-subgraph. Fowler [25] has shown that there are at least two 4-colourings of plane triangulations without vertices of degree 3. A bridgeless cubic planar graph can be face-4-coloured if and only if it can be properly edge-3-coloured (a Tait colouring). The dual of a cubic graph is a triangulation. A simple triangulation has a Grünbaum colouring if it can be 3-edge-coloured so that each colour appears around each face. Grünbaum [30] conjectured that: Any simple triangulation of an orientable surface can be 3-edge- coloured so that each colour appears around each face. The conjecture was false: Kochol recently announced that he has found a simple triangulation on the 9-holed torus that does not have a Grünbaum colouring. Nonetheless, several interesting questions remain: Question 5.3 Does every triangulation of the torus have a Grünbaum colouring? A ‘yes’ answer to this question should be difﬁcult to prove, since it is strictly stronger than the four-colour theorem: any planar triangulation could be placed in a single face of any triangulation of the torus, and a Grünbaum colouring of the resulting triangulation would give the dual of a Tait colouring of the planar graph. Question 5.4 Does every triangulation of a surface by a complete graph have a Grünbaum colouring?

Flows Tutte [85] made a famous sequence of conjectures that we now describe. Let G be a directed graph.Ak-flow on G is a function f : E(G) →{0, ±1, ±2,...,±(k−1)} such that, at each vertex v, the sum of the flows on the arcs directed into v is equal to the sum of the flows on the arcs directed out of v. The flow is nowhere-zero if no arc gets flow 0. It is easy to show that the existence of a nowhere-zero k-flow on a graph is independent of the directions assigned to the edge – if you reverse the direction on an edge, just negate its flow. Conjecture 5.5 Every 2-edge-connected graph has a nowhere-zero 5-flow. Conjecture 5.6 Every 2-edge-connected graph with no Petersen minor has a nowhere-zero 4-flow. Conjecture 5.7 Every 4-edge-connected graph has a nowhere-zero 3-flow. 324 Dan Archdeacon

Seymour [74] proved that every bridgeless graph has a nowhere-zero 6-ﬂow. Any k-ﬂow must assign 0 to every cut-edge, so the assumption that the graph be 2-edge-connected is necessary. Conjecture 5.6 has been proved for cubic graphs by Robertson, Sanders, Seymour and Thomas (in preparation). For more details on these conjectures, see Zhang [95].

6. Local planarity

Planar graphs have many interesting properties. If a graph is embedded on a surface so that large neighbourhoods of a vertex are embedded in a planar fashion, does it share these properties? We make the above more precise. Let G be a graph embedded on a surface S. The face-width fw(G) of the embedding, also known as the representativity ρ(G), is the minimum value of |C ∩G|, taken over all non-contractible cycles C in the surface. Equivalently, it is the smallest k such that there is a sequence of (closed) faces f1,f2,...,fk with a non-contractible cycle contained in f1 ∪ f2 ∪···∪fk. The edge-width ew(G) is the graph-theoretic length of the shortest non-contractible cycle in the embedded graph G. Note that fw(G) ≤ ew(G), with equality for triangulations. These parameters measure the degree of ‘local planarity’ of an embedding.

Colouring problems The ﬁrst property that we might be interested in for locally planar graphs is 4-colourability. Fisk [24] showed that there are triangulations of arbitrarily large face-width that are not 4-chromatic. Thomassen [83] showed that if G is embedded in a surface of genus g so that every non-contractible cycle has length at least 214g+6, then G is 5-colourable. The following related dual question is due to Robertson:

Question 6.1 Is there a number k such that any 2-connected cubic graph embedded on a surface with face-width at least k is 3-edge-colourable?

This question is related to Grünbaum colourings – see Question 5.3.

Hamiltonian cycles

In Section 4 we discussed conditions on connectivity that imply that a graph embedded on a surface is Hamiltonian. We now add a local planarity condition to the embedding. 15 Open problems 325

Conjecture 6.2 There is a function f(g) such that every 5-connected graph embedded on a surface of genus g with face-width at least f(g) contains a Hamiltonian cycle.

This conjecture is true for triangulations by a result of Yu [93], but it does not extend to 4-connected graphs, even triangulations, as noted by Thomassen [84]. The function f(g)cannot be constant [5].

Separating cycles One nice feature of graphs embedded with large face-width is the existence of subgraphs with certain properties. In fact, for any graph H embedded on a surface S, there is a constant k(H) such that each graph embedded on S with face-width at least k(H) contains H as a surface minor (see [67] or [53]). This includes the planarizing cycles of Thomassen [83]. The following problems involve non- contractible separating cycles. The ﬁrst is due to Barnette:

Conjecture 6.3 Every triangulation of an orientable surface of genus at least 2 contains a non-contractible surface-separating cycle.

Zha and Zhao [94] showed that this conjecture is true for triangulations of face-width at least 6. Zha has made the following stronger conjecture.

Conjecture 6.4 Every graph embedded on an orientable surface of genus at least 2 with face-width at least 3 contains a non-contractible surface-separating cycle.

Brunet, Mohar and Richter [14] have shown that Conjecture 6.4 is true if the face-width is at least 6. Finally, Mohar [51] reports that Thomassen made the following even stronger conjecture:

Conjecture 6.5 Let G be a triangulation of a surface of genus g, and let 1 ≤ h

This conjecture may even be true for an arbitrary embedded graph of face-width at least 3.

7. Thickness, book embeddings and covering graphs

Sometimes representing a non-planar graph in the plane without crossings is essential. We examine various ways to do so without using surfaces of higher genus. 326 Dan Archdeacon

Thickness The thickness t(G) of a graph G is the minimum number t such that G has t planar subgraphs H1,H2,...,Ht that collectively contain each edge of G. The thickness of the complete graph is known (see [89] for a history of this result). The thickness of the complete bipartite graph is unknown. The best result is due to Beineke, Harary and Moon [7], who showed that t(Kr,s ) =rs/2(r + s − 2), except possibly when r and s are both odd and, with s ≤ r, there is an even integer k with r =k(s − 2)/s − k. The exceptional cases are quite rare.

Problem 7.1 Find the thickness of Kr,s for all r, s.

Consider the case where

• every edge in each subgraph is a line segment; • every vertex is represented by the same point in each drawing.

The resulting minimum number of subgraphs is called the geometrical thickness or linear thickness of G. This is equivalent to the minimum number of colours k such that a rectilinear drawing of G can be k-edge-coloured with no two edges of the same colour crossing.

Problem 7.2 Find the geometrical thickness of the complete graph.

To my knowledge, the oldest work on this problem is in unpublished notes 1 n K of Grinberg, who gave an upper bound of 4 for n. This bound was also discovered by Dillencourt, Eppstein and Hirshberg [20], who also gave a lower bound of (n/5.646) + 0.342. They related geometrical thickness with the page number of a graph (see the material preceding Problem 7.5).

Earth–Moon colourings We next consider maps on two spheres – say, the Earth and the Moon. Each simply connected region on the Earth is a country and has a corresponding simply connected colony on the Moon. We desire to colour the countries and colonies so that:

• each country receives the same colour as its colony; • if two countries or colonies share a common boundary edge, then they receive distinct colours.

Ringel [65] asked for the following Earth–Moon colouring number.

Question 7.3 What is the minimum number of colours necessary to colour all Earth–Moon maps? 15 Open problems 327

Ringel noted that this number is between 8 and 12. The lower bound is easily demonstrated by drawing 8 states, any one adjacent to any other on either the Earth or the Moon. The upper bound follows from Euler’s formula. The dual form of the problem asks for the maximum chromatic number among all graphs of thickness 2. The complete graph K8 is of thickness 2, but K9 has thickness 3. As reported in [28], Sulanke showed that K11 − C5 is of thickness 2 and has chromatic number 9, so the Earth–Moon colouring number lies between 9 and 12.

Question 7.4 What is the maximum chromatic number among all graphs of thickness t?

An upper bound of 6t is given by Euler’s formula. The known formula for the thickness of the complete graph gives a lower bound of 6t − 2 when t ≥ 3 (see [89]). Hutchinson wrote an excellent paper [39] including this topic.

Book embeddings A page is a closed half-plane, and a book is a collection of pages identiﬁed along their boundaries; this common boundary is called the spine. A book provides another topological space in which to embed graphs. Each graph embeds in a book with three pages, so it is usual to put some restrictions on how the graph is drawn. Speciﬁcally, we require that all vertices lie on the spine and no edge contains a point on the spine other than its ends. The page number pn(G) of a graph G is the fewest number of pages in a book containing G. Book embeddings were introduced by Kainen (see [44], [8]). Chung, Leighton and Rosenberg [16] gave some interesting applications of book embeddings to VLSI design. K 1 n The page number of the complete graph n is 2 . Surprisingly, the page number of the complete bipartite graph Kr,s is much harder.

Problem 7.5 Find the page number of Kr,s . 1 ( r + s) The upper bound conjectured in [54] is 4 2 . Recently Enomoto et al. [21] found some drawings that use fewer pages. In particular, they can draw Kr,r in r/ + K n − only 2 3 1 pages and they can draw n2/4,n in 1 pages. Finally, they give the asymptotic result

{m (K ) = n}= 1 n2 ( + o( )) . min :pn m,n 4 1 1

Planar covers Agraph G covers a graph H if there exists a graph map f from G to H such that the edges incident with a vertex v map bijectively to those incident with w = f(v); 328 Dan Archdeacon the set f −1(w) is called the fibre above w.IfH is connected, then the size of each fibre is a constant, called the fold number of the covering. For example, the dodecahedron is a 2-fold cover of the Petersen graph; the fibre above a vertex of the Petersen graph consists of two vertices at distance 5 in the dodecahedron. More strongly, taking the usual embedding of the dodecahedron in the sphere and identifying all spherical antipodal points yields an embedding of the Petersen graph in the projective plane. Each graph H that embeds in the projective plane has a planar covering graph G created by the spherical antipodal double covering of the projective plane, described above. The following conjecture by Negami [56] states that this sufficient condition is also necessary.

Conjecture 7.6 A graph has a ﬁnite planar cover if and only if it embeds in the projective plane; in particular, K1,2,2,2 has no ﬁnite planar cover.

It follows that if a graph has a planar cover, then it has a 2-fold planar cover. Hence the minimum fold number among all planar covers is 1, 2 or ∞, so this is sometimes called the 1-2-∞ conjecture. Note that the property of having a planar cover is hereditary under the minor ordering. So, to prove the conjecture it sufﬁces to show that the 35 minor-minimal non-projective-planar graphs (see [4]) have no planar covers. Some of these 35 cases have been done directly, while some reduce to others (see [38]). If K1,2,2,2 has no planar cover, then the general conjecture is true.

8. Geometrical topics

There is a rich set of problems concerning representations of planar graphs. For example, what is commonly known as Fáry’s theorem [22], but was earlier proved by Wagner [87], asserts that every planar graph has an embedding in the plane for which each edge is a straight-line segment. These are also known in the literature as geometrical graphs or as rectilinear embeddings.

Integer-length edges Harborth asks the following (see [34], [35]):

Question 8.1 Is it possible to draw every planar graph so that each edge is a line segment of integer length?

Every graph has an embedding in Euclidean 3-space so that each edge is a line segment. A common construction is to place the tth vertex at the point (t, t2,t3) and represent edges by line segments. 15 Open problems 329

Question 8.2 Does every graph have an embedding in 3-space so that the edges are line segments of integer lengths? If the answer to either of the preceding two questions is no, then what can be said about the minimum number d such that the graph has an integer-length representation in Euclidean d-space?

Intersecting line segments Let L be a collection of line segments in the plane. Form a graph G = G(L) whose vertices are the segments in L and whose edges join those line segments that cross each other. We say that G is represented by L. Question 8.3 Is every planar graph G representable by a set of line segments? The problem was introduced by Scheinerman [71], who proved that every outerplanar graph is so representable. A stronger question is due to West [88]: Question 8.4 Is every planar graph representable by intersecting line segments with four different slopes? Since this implies the four-colour theorem, any proof is expected to be difﬁcult. Hartman, Newman and Ziv [36] proved that every bipartite planar graph is so representable, using segments with only two slopes. Scheinerman asked the following two questions about representing three- colourable graphs: Question 8.5 Is every 3-colourable planar graph representable by intersecting line segments, regardless of slopes? Question 8.6 Is every 3-colourable planar graph representable by intersecting line segments with three different slopes?

Discs and spheres

Let D1,D2,...,Dn be a set of discs in the plane whose interiors are pairwise disjoint. Form a graph G(D) whose vertices are the Di, with an edge joining Di and Dj if and only if the boundaries of the discs touch. We say that the discs represent G. Koebe [47] proved that every planar graph can be realized in the plane by by a set of discs. Question 8.7 Which graphs can be represented by spheres in 3-space?

The structure of these graphs is interesting. Neil Sloan says that K6 is not in this class, but K5 is. 330 Dan Archdeacon

Question 8.8 Does there exist a ‘Kuratowski-type’ forbidden subgraph characterization of graphs that can be represented by spheres in 3-dimensional space?

Consider a set of n points in the plane. Around each point i ﬁnd a disc Di that contains none of the other n points. Form a graph G by considering the points as vertices, with an edge joining i and j if and only if their discs intersect. Call this an overlapping disc representation of the graph (these are also called sphere of inﬂuence graphs). Problem 8.9 Characterize those graphs with overlapping disc representations. Does K9 have such a representation?

Intersecting curves Consider n simple closed curves in the plane for which any two curves have at least one point in common, but no point lies on three or more curves. The curves need not cross transversally. The minimum number of intersections over all such families of curves is denoted by f (n). f (n) f (n)/ n = Problem 8.10 Find .Islimn→∞ 2 2? n f (n) ≤ n n There must be at least 2 points of intersection. Also, 2 2 – consider circles with the same radius but with centres perturbed by εi from the origin. It is known that f(4) = 6, f(5) = 12 and f(6) = 20; the smallest unknown value is f(7). Richter and Thomassen [63] give some background on this problem.

9. Algorithms

We now turn our attention to polynomial-time algorithms for ﬁnding parameters of graphs and embeddings.

Finding the genus It is known that determining the orientable or non-orientable genus of a graph is NP-complete (see [82]). Fiedler, Huneke, Richter and Robertson [23] found an easily calculated formula for the orientable genus of projective-planar graphs, and Robertson and Thomas [70] found a similar result for graphs embeddable on the Klein bottle. It follows that there is a polynomial-time algorithm for finding the orientable genus of these graphs. The following conjecture (see [70], [23]) asserts that this holds for any fixed non-orientable surface. Conjecture 9.1 For each fixed k, there is a polynomial-time algorithm to calculate the orientable genus of a graph with non-orientable genus k. 15 Open problems 331

A family of graphs is minor-closed if every minor of a graph in the family is also in the family. Robertson asked the following question: Question 9.2 For which minor-closed families of graphs can the genus be calculated in polynomial time? The graph minor theorem of Robertson and Seymour [68], [69] gives a polynomial-time algorithm for testing membership in such a family. Mohar has told me that for some families, such as those of bounded tree-width, there is a polynomial-time algorithm. For others, such as apex graphs (those graphs with a vertex whose deletion yields a planar graph), calculating the genus is NP-complete [52].

Finding cycles We next turn our attention to a graph G embedded on a surface. Thomassen [81] showed that there are polynomial-time algorithms for finding the shortest non- contractible cycle in G, the shortest non-separating cycle in G, and the shortest orientation-reversing cycle in the embedded G. Question 9.3 Let G be embedded on a surface. Do there exist polynomial-time algorithms to find: • a shortest contractible cycle in G? • a shortest surface-separating cycle in G? • a shortest orientation-preserving cycle in G? The first problem is easy if G has no vertices of degree 2 or less (see Thomassen [81]).

Finding LEW-weight functions A large edge-width (or LEW) embedding is one where the edge-width is larger than the maximum length of a facial walk. Thomassen [81] showed that there is a polynomial-time algorithm that, given a 3-connected graph G, constructs an LEW- embedding of G or concludes that no such embedding exists. Suppose that each edge receives a positive weight. The weight of a walk is the sum of the weights of its edges, counting multiplicity. A LEW-weight function is a weighting function for which the weight of any non-contractible cycle is greater than the weight of every facial walk. The following questions appear in [53]: Problem 9.4 Do there exist polynomial-time algorithms for the following questions? • Given an embedded graph G, does G have an LEW-weight function? • Does a given graph G have an embedding with an LEW-weight function? 332 Dan Archdeacon

10. Infinite graphs Colouring the plane The most famous open problem connecting infinite graphs and the plane concerns the chromatic number of the unit distance graph, in which the points in the Cartesian plane are the vertices, and two are adjacent if they are at distance 1. An example shows that the chromatic number is at least 4, while a slight variation on the hexagonal lattice shows that it is at most 7 (see [80]). Problem 10.1 Find the chromatic number of the unit distance graph. Of course, improving either bound would be significant.

Geodesics A geodesic in an infinite graph G is a doubly-infinite path D such that the distance between any two vertices in D is the same as their distance in G. Bonnington, Imrich and Watkins [12] conjecture the following: Conjecture 10.2 Let G be an infinite, locally-finite, 1-ended vertex-transitive graph. Then G is planar if and only if for every geodesic D, G − V(D) has exactly two components, both infinite.

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DanArchdeacon [[email protected]] received his Ph.D. degree fromThe Ohio State University. He is currently a Professor of Mathematics and Statistics at The University of Vermont, where he has been for over twenty-ﬁve years. He was recently named a University Scholar in recognition of his contributions to discrete mathematics. His research interests focus on topological graph theory, but include all of graph theory, design theory, and combinatorics in general. For many years he was the primary Managing Editor of the Journal of Graph Theory. Lowell Beineke [[email protected]] is Schrey Professor of Mathematics at Indiana University–Purdue University Fort Wayne, where he has been since receiving his Ph.D. degree from the University of Michigan under the guidance of Frank Harary. His graph theory interests are broad, and include topological graph theory, line graphs, tournaments, decompositions and vulnerability. With Robin Wilson he edited Selected Topics in Graph Theory (3 volumes), Applications of Graph Theory, Graph Connections and Topics in Algebraic Graph Theory. Until recently he was editor of the College Mathematics Journal. Jianer Chen [[email protected]] received a Ph.D. degree in Computer Science from the Courant Institute of Mathematical Sciences, New York University, in 1987, and a Ph.D. degree in Mathematics from Columbia University in 1990. He is currently a Professor of Computer Science at Texas A & M University. His research interests include graph theory and algorithms, computational optimization and complexity theory, computer graphics, computer networks and bioinformatics. Mike Grannell [[email protected]] received his Ph.D. degree from Imperial College, London, in 1970. From 1974 to 1997 he worked at what is now the University of Central Lancashire, where he started his long-term collaboration with Terry Griggs. Since then he has been a Professor of Mathematics at The Open University, UK. His research interests are mainly in combinatorial designs, including conﬁgurations and topological representations of designs. He has published around one hundred papers on combinatorial mathematics.

337 338 Notes on contributors

Terry Griggs [[email protected]] is a Professor of Mathematics at The Open University, UK. He started his long-term collaboration with Mike Grannell in 1977 when both were at the University of Central Lancashire in Preston, UK. His research interests are mainly in combinatorial designs, in particular their topological representations and the history of their development. He has published around one hundred papers on combinatorial mathematics. Jonathan Gross [[email protected]] is Professor of Computer Science at Columbia University. His mathematical work in topology and graph theory have earned him an Alfred P. Sloan Fellowship, an IBM Postdoctoral Fellowship, and numerous research grants. With Thomas Tucker, he wrote Topological Graph Theory and several fundamental pioneering papers on voltage graphs and on enumerative methods. He has written and edited eight books on graph theory and combinatorics, seven books on computer programming topics, and one book on cultural sociometry. Yuanqiu Huang [[email protected]] earned his Ph.D. degree in 1996 at the Institute ofApplied Mathematics,Academia Sinica, Beijing, China. He is currently a Professor of Mathematics at the Hunan Normal University, China, where he has been since 1996. His main research interests lie in topological graph theory, mathematical combinatorics and optimization. Joan P. Hutchinson [[email protected]] earned her Ph.D. degree in 1973 at the University of Pennsylvania, under the supervision of H. S. Wilf. After a John Wesley Young Research Instructorship at Dartmouth College, she spent about half of her career at Smith College and half at Macalester College. She is now retired from teaching, continues research in chromatic, topological and visibility graph theory, and would like to solve some of the open problems of Chapter 6. She was a 1994 recipient of a Carl. B. Allendoerfer Award for an article in Mathematics Magazine. She won a 1998 MAA North Central Section Teaching award and a 1999 Deborah and Franklin Tepper Haimo Award for Excellence in College or University Teaching. Jin Ho Kwak [[email protected]] is a Professor in the Department of Mathematics at the Pohang University of Science and Technology in Korea, and is the director of the Combinatorial and Computational Mathematics Center. He works on combinatorics and topology, mainly on covering enumeration related to Hurwitz problems and regular maps on surfaces. He has published over one hundred papers on these topics. Jaeun Lee [[email protected]] is a Professor in the Department of Mathematics at the Yeungnam University in Korea, and is a faculty member of the Combinatorial and Computational Mathematics Center. He works on combinatorics and topology, mainly on enumerations of graph coverings and bundles. Notes on contributors 339

Bojan Mohar [[email protected]] is a Professor at Simon Fraser University, Canada, and at the University of Ljubljana, Slovenia. He has published over 180 research papers, many of which are in topological graph theory. His other research interests include graph minors, graph colouring, graph algorithms and algebraic graph theory. He is a coauthor of a monograph, Graphs on Surfaces. In 1990 he was awarded the Boris Kidricˇ prize, the Slovenian national award for exceptional achievement in science. Currently, he holds the position of a Canadian Research Chair in Graph Theory. He is an elected member of the Engineering Academy of Slovenia. Tomaž Pisanski [[email protected]] studied mathematics and computer science in Slovenia, France and the USA. In 1981 he received his Ph.D. degree from the University of Ljubljana, Slovenia, under the guidance of Torrence Parsons, where he is now a Professor of Mathematics with teaching experience gained in Austria, Canada, Croatia, Italy, New Zealand, Slovenia and the USA. He was the Neil Grabois visiting professor at Colgate University (2003). He is the initiator of graph theory in Slovenia with research interests connecting discrete mathematics to other branches of mathematics and applications in the sciences. He is a founding editor of the Ars Mathematica Contemporanea and Vice-President of the International Academy for Mathematical Chemistry. Bruce Richter [[email protected]] is a Professor of Mathematics at the University of Waterloo, where he has been since 1999. Previously, he was at Carleton University, the U. S. NavalAcademy, The Ohio State University and Utah State University. His research interests have centred around the interplay between topology and graph theory, focusing on embeddings of graphs in surfaces, crossing numbers of graphs and, more recently, topological spaces associated with infinite graphs. Gelasio Salazar [[email protected]] is a Professor at the Institute of Physics at the Universidad Autónoma de San Luis Potosi (Mexico), where he has been since 1997. He has held visiting positions at the Georgia Institute of Technology and The Ohio State University. His main research interests lie in topological graph theory, discrete geometry, and discrete dynamical systems. Jozef Širá˘n [[email protected]] is a Professor of Pure Mathematics at The Open University, UK, and a Professor of Mathematics at the Slovak University of Technology, Slovakia. His research interests have focused on the interplay between graphs, groups and surfaces, with emphasis on graph embeddings with high level of symmetry. Thomas Tucker [[email protected]] is the Charles Hetherington Professor of Mathematics at Colgate University, where he has been since 1973, after obtaining his Ph.D. degree in 3-manifolds from Dartmouth in 1971 and taking 340 Notes on contributors up a post-doctoral position at Princeton University (where his father A. W. Tucker was chairman and John Nash’s thesis advisor). He is co-author (with Jonathan Gross) of Topological Graph Theory. His early publications were on non-compact 3-manifolds, then topological graph theory, but his recent work is mostly algebraic, especially distinguishability and the group-theoretic structure of symmetric maps. Mark Watkins [[email protected]] earned his A.B. at Amherst College and his Ph.D. degree (1964) at Yale University under the supervision of Øystein Ore. He taught at the University of North Carolina / Chapel Hill and the University of Waterloo before joining the mathematics department at Syracuse University in 1968. He has held visiting positions at the Technical University of Vienna and the Université de Paris-Sud. He has published over sixty research articles, mainly in algebraic and topological graph theory; his Erdos˝ number is 2, via six co-authors. He is also co-author (with J. E. Graver) of a graduate combinatorics text and a definitive classification of edge-transitive graphs. Arthur White [[email protected]] is a Professor of Mathematics at Western Michigan University, where he has been (with the exception of four sabbatical years in England) since receiving his Ph.D. degree from Michigan State University in 1969. In 1996 he received an MAAMichigan Section teaching award. He authored Graphs, Groups, and Surfaces (two editions) and Graphs of Groups on Surfaces. His work in topological graph theory has emphasized the genus of graphs, genus of groups, maximum genus, dual maps, symmetric maps, embeddings and designs, enumerative topological graph theory, random topological graph theory, English church-bell ringing, modelling finite fields and, most recently, modelling finite geometries. Robin Wilson [[email protected]] is a Professor of Pure Mathematics at The Open University, UK, and Emeritus Professor of Geometry at Gresham College, London. After graduating from Oxford, he received his Ph.D. degree in number theory from the University of Pennsylvania. His research interests include graph colourings and the history of combinatorics, and he has Erdos˝ number 1. He has written and edited many books on graph theory and the history of mathematics, including Introduction to Graph Theory and Four Colours Suffice, and has won Lester Ford and George Pólya Awards from the MAA for his expository writing. Arjana Žitnik [[email protected]] received her Ph.D. degree in 2002 from the University of Ljubljana, Slovenia, under the supervision of Tomaž Pisanski. She is an Assistant Professor there and is also a researcher at the Institute of Mathematics, Physics and Mechanics in Ljubljana. Her main research activity is in discrete mathematics with applications, in particular in topological graph theory, graph algorithms, algebraic combinatorics and configurations. She is also interested in cryptology and coding theory. Index

absolute Galois group, 222 blocks of a transversal design, 269 accumulation point, 304 blocks of a triple system, 269 action of a group, 24, 226 book embedding, 327 adjacent edges,2 boundary walk, 20 adjacent vertices,2 bounded automorphism, 294 afﬁne plane, 253 bouquet, 5, 50 algorithms, 30, 73, 330 bow-tie operation, 72 almost-4-connected graph, 306 branch decomposition, 84 almost homocyclic group, 216 branch set, 191 almost planar graph, 144 branched covering, 26, 191 almost-transitive graph, 293 bridge, 9, 89 angle transformation, 169 Brooks’s theorem,9 apex graph, 76, 331 Burnside’s lemma, 184 arc, 10, 200 arc-transitive graph,2 atom, 297 cactus, 38 attach an ear, 58 cage, 252 attach serially, 58 Cartesian product,6 automorphism, 2, 22 Cayley graph, 28, 225 automorphism group,2 Cayley map, 29, 210, 227 automorphism of a map, 201, 305 C-disc embedding, 68 average crossing number, 143 cellular embedding, 14, 20 average genus, 56 chiral map, 206 choosable graph, 126 chromatic index,9 balanced automorphism, 213 chromatic number of a graph,9 balanced Cayley map, 211 chromatic number of a map, 22 balanced representation, 155 chromatic number of a surface, 16, 22 bar, 46 circle packing representation, 66 bar-amalgamation, 46 circuit,3 barycentric representation, 158 circular embedding, 20 barycentric subdivision, 169 circular embedding conjecture, 20, 320 base graph, 27, 182 closed surface, 19 bead of a necklace, 38 closed walk,3 biembeddable pair, 271 closed-end ladder, 48 biembedding, 271 cobblestone path, 5, 49 Bilinski diagram, 307 co-degree of a face, 304 bipartite graph,3 colony, 326 bisection width, 138 colouring a graph, 9, 111 blockage, 77 combinatorial conﬁguration, 174 blocking curves, 77 combinatorial current graph, 26

341 342 Index

complement, 2 Desarguian plane, 250 complete bipartite graph,5 d-fibre, 292 complete circuit, 282 diagonal curve, 122 complete graph,4 diameter,3 complete k-partite graph,5 digraph, 10 component,3 dilation coefficient, 154 concentric Bilinski diagram, 207 dipole, 5, 51 configuration embedding, 256 directed edge, 10 congestion of an edge, 138 directed graph, 10 congruent embeddings, 53 direction, 19 connected component,3 disc, 175 connected covering, 185 disc embedding, 68 connected graph,3 disc representation, 329 connectivity,9 disconnected graph,3 connectivity of an infinite graph, 296 disjoint crossing paths, 68 contractible,8 displayed sets, 85 contraction of an edge,8 distance, 3, 118 convex representation, 165 distribution of inverses, 211 core-free permutation group, 202 dominated end, 291 co-tree, 36 double ray, 290 covering, 26, 182 drawing of a graph, 135, 314 covering digraph, 27 duad, 262 covering graph, 27, 182 dual edge, 21 covering isomorphism, 182 dual embedding, 21 covering projection, 26, 181 dual map, 201 covering transformation, 182 dual vertex, 21 covering transformation group, 182 dual-width of an embedding, 146 cover, 327 Coxeter graph, 207 E-adjacent embeddings, 59 Cremona–Richmond geometry, 262 ear decomposition, 35 critical graph, 114 Earth–Moon colouring, 326 crosscap, 12 edge,1 crosscap distribution polynomial, 47 edge fibre, 28 crosscap number, 16, 20, 47 edge-accumulation point, 303 crosscap range, 20, 47 edge-connectivity,9 crossing number, 11, 23, 133, 136, 314 edge-homogeneous map, 308 crossing-critical graph, 139 edge-set,1 crossing-free curves, 77 edge-symbol, 308 cubic graph,2 edge-transitive graph,2 current graph, 26, 281 edge-width, 324 cut-edge,9 eigenpolytope, 177 cut-vertex,8 eigenvector method, 156 cycle,3 element of a geometry, 174 cycle double cover conjecture, 20, 320 elementary Abelian p-group, 216 cycle graph,4 embeddable, 14 cycle rank, 34 embedding, 10, 14, 20, 153 cycle type, 309 embedding extension problem, 74 embedding a grid, 107 embedding in the plane, 10 dart group, 24, 200 embedding of a triple system, 270 decomposition of a graph, 84 embedding theory, 22 deficiency of a graph, 36 end vertex,2 degree of a vertex, 2, 19 endpoint of a curve, 175 deletion of a vertex,7 endpoint of an edge, 19 deletion of an edge,8 end of an ear, 58 depth of trap, 59 energy function, 155 d-equivalence, 292 enumeration, 29 Desargues geometry, 258 equivalence, 292 Index 343

equivalent representations, 154 geometrical crossing number, 316 equivalent rotation systems, 24 geometrical graph, 328 Euler characteristic, 14, 21 geometrical thickness, 326 Euler genus, 64, 107, 226 girth of a graph,3 Euler genus problem, 75 good drawing, 171, 314 Euler’s formula, 11, 14, 21 graph, 1, 19 Eulerian graph,3 graph drawing, 63, 151 Eulerian trail,3 graph minor, 31 even component, 36 Graph Minors Project, 81 evenly embedded graph, 115 graph nodal domain theorem, 157 excluded minor, 69 graph on a surface, 14 exponential growth, 300 graph representation, 153 extended edge-symbol, 310 grid, 82 Grötzsch’s theorem, 124 group of a transversal design, 269 face, 11, 14, 20, 164 growth degree, 300 face boundary walk, 20 Grünbaum colouring, 323 face-transitive map, 305 face-width, 117, 324 factorization, 213 Hajós join, 114 faithful action, 24 Halin graph, 322 Fano plane, 174, 245 Hamiltonian graph,3 Fáry’s theorem, 152 handle, 12 fibre, 28, 292, 328 Heawood map colour problem, 23, finite projective plane, 247, 250 63, 112 finite-connectivity, 295 Heawood number, 23 flag, 25, 163, 174, 203, 264 Heffter’s first difference problem, 282 flag graph, 165 hereditary, 318 flag-simple, 164 high-end colouring, 113 flat type, 24 homeomorphic graphs,8 flow, 323 homeomorphically irreducible tree, 322 F -map, 205 homogeneous representation, 176 fold number, 328 Hurwitz system, 191 forbidden family, 17 Hurwitz’s theorem, 232 forbidden subgraph, 69 force-directed placement, 160 forest,4 four-colour theorem, 22, 322 immersion of a graph, 138 fragment, 297 incidence geometry, 174 framework, 177 incidence relation, 174 fullerene, 156 incidence structure, 174 fundamental polygon, 172 incident vertex and edge,2 independence ratio, 117 independent diagonal curve, 122 general rotation system, 24 independent odd crossing number, 136 generalized Laplacian, 156 induced subgraph,7 generalized Petersen graph, 143 infinite-connectivity, 296 generalized pseudosurface, 248 inserting an edge, 35 generalized quadrangle, 262 integer-length edge, 328 generic iterative graph representation algorithm, intermediate growth, 300 162 intersecting line segments, 329 genus, 15, 20 irredundant generating set, 236 genus distribution polynomial, 46 isolated vertex,2 genus of a geometry, 249 isomorphic branched coverings, 191 genus of a group, 28, 225 isomorphic coverings, 182 genus problem, 75 isomorphic graphs, 2, 22 geodesic, 332 isomorphism of embeddings, 22 geodetic edge, 306 isomorphism of graphs, 2, 22 344 Index

join, 2 locally planar embedding, 113 join of graphs,6 loop,1 JW group, 241 low-end colouring, 116 k-choosable graph, 126 map, 163 k-chromatic graph,9 map automorphism, 201, 305 k-colourable graph,9 map colour problem, 317 k-colour-critical graph, 114 map colour theorem, 16 k-connected graph, 9, 100 map on a surface, 22 k-crossing-critical graph, 139 map theory, 22 k-degenerate graph, 126 Maschke’s characterization, 235 k-dimensional tree,4 maximum crosscap number, 20, 47 k-edge-colourable graph,9 maximum genus, 20, 34, 46 k-edge-connected graph,9 medial map, 167, 221 k-flow, 323 Mendelsohn triple system, 260, 270 K-graph, 71 Menger graph, 248 k-grid, 82 Menger’s theorem,9 meridian, 122 Kirchhoff’s current law, 281 M Kirkman triple system, 260 -flag simple map, 164 k-list-colourable graph, 126 minimal forbidden minor, 69 k-regular graph,2 minimal forbidden subgraph, 69 k-separation, 69 minimum crosscap number, 20, 47 k-tree,4 minimum genus, 20, 46 Klein bottle, 12 minor, 8, 31, 69, 82 minor ordering, 318 Kuratowski’s theorem, 11, 64 minor-closed family, 331 Kuratowski-type theorem, 38 minor-minimal forbidden family, 17 mixed type, 309 Möbius function, 185 ladder, 48 Möbius strip, 12 Laplace method, 156 monodromy group, 24, 25, 200 Laplacian, 156 morphism of maps, 201 large edge-width embedding, 331 multiple edges,1 lattice, 301 leapfrog transformation, 169 ledge numbers, 216 natural projection, 28 left action of a group, 24 necklace, 5, 38, 56 left regular representation, 182 negative support, 157 length of walk,3 neighbour,2 Levi graph, 165, 174, 248 neighbourhood, 2, 181 LEW embedding, 331 neighbourhood graph, 270 LEW weight function, 331 neighbourhood system, 292 lexicographic product,6 never-increasing sequence, 95 lifting an embedding, 29 non-degenerate representation, 153 line of an incidence structure, 174 non-orientable genus, 16, 226, 317 lineal incidence structure, 174 non-orientable genus distribution, 47 linear crossing number, 316 non-orientable genus problem, 75 linear drawing, 316 non-orientable surface, 12 linear genus, 172 non-separable graph,8 linear growth, 300 normalized assignment, 183 linear thickness, 326 nowhere-zero flow, 323 linked decomposition, 89 number of embeddings, 46, 47 linked edges, 94 list-colourable graph, 126 local rotation, 315 obstruction, 63, 318 locally bipartite embedded odd component, 36 graph, 115 odd crossing number, 136 locally finite graph, 289 one-way infinite path, 290 Index 345

oppositely oriented map, 201 polynomial growth, 300 orbit-counting theorem, 184 positive support, 157 order of a blockage, 77 PQ-tree, 65 order of a graph, 1 presentation, 29 order of a projective plane, 250 primal embedding, 21 ordinary derived graph, 182 projective plane, 12, 176, 250 ordinary voltage assignment, 182 proper minor, 82 orientable genus, 317 proper separation, 69 orientable genus range, 20, 46 pseudosurface, 248, 319 orientable surface, 12, 19 pure, 52 orientation-preserving automorphism 24, 201 orientation-reversing automorphism, 24, 201 oriented map, 200 quadrangulation, 14, 116 oriented surface, 19 quadratic growth, 300 orthonormal representation, 155 quasi-geodetic ray, 292 outer-embeddable graph, 67 outerplanar graph, 66 overlap matrix, 52 rank of a graph, 302 ray, 290 ray type, 309 real projective plane, 176 page, 327 rectilinear crossing number, 145, 316 page number, 327 rectilinear drawing, 316 pair crossing number, 136 rectilinear embedding, 328 Pajek program, 160 reflection, 205 Pappus graph, 176 reflexible, 205 Pappus’s geometry, 262 region, 10, 14, 20 partite sets,4 regular action, 24 Pasch configuration, 262 regular Cayley map, 212 path,3 regular covering, 26, 182 path graph,4 regular embedding, 264 perfect matching, 321 regular graph, 2, 19 periodic graph, 302 regular map, 205 peripheral cycle, 158 regular orientable map, 202 permutation derived graph, 182 regular representation, 182 permutation voltage assignment, 182 regular surface branched covering, 193 permutation voltage graph, 27 regular voltage assignment, 27 Petersen colouring conjecture, 321 regular voltage graph, 27 Petrie cycle, 218 relator, 29 Petrie dual, 218 representation, 153 Petrie path, 218 representation of a geometry, 175 Petrie walk, 308 representation polytope, 177 Petrie-orbits, 309 representativity, 117, 146, 324 piecewise-linear structure, 19 r-fold covering, 182 pinch point, 319 Riemann–Hurwitz equation, 229 planar cover, 327 Ringel’s theorem, 16 planar dilation coefficient, 154 Ringel–Youngs theorem, 15 planar graph, 10 Robertson–Seymour theorem, 17, 70 planar map, 303 root of Bilinski diagram, 307 planar representation, 153 rooted forest, 95 planar tile, 141 rooted minor, 96 planarity problem, 22 rotation, 23, 315 planarizing set, 118 rotation scheme, 23 plane graph, 10 rotation system, 23 plane representation, 153 rotational projection, 170 r Poincaré dual embedding, 21 R -representation, 153 point of incidence structure, 174 points of a transversal design, 269 points of a triple system, 269 Schlegel drawing, 159 346 Index

Schlegel representation, 159 tetrahemihexahedron, 166 semi-edge, 200 theory of algorithms, 63 separating cycles, 325 thick end, 291 separator, 296 thickness of a graph, 11, 326 serial attachment, 58 thin end, 291 set of planarizing cycles, 118 thrackle, 316 sheet of a spindle, 319 thrackle conjecture, 316 sign, 24 tile, 141 signature, 231 tile crossing number, 141 simple curve, 175 tile drawing, 141 simple graph,1 T -normalized assignment, 183 simple triple system, 269 topological end, 293 sink of an ordering, 64 topological model, 248 size of a face, 11 topological ordering, 318 skeleton, 164 torso, 69 skew morphism, 217 total embedding distribution, 51 source of an ordering, 64 traceable graph,3 S-outer-embeddable graph, 67 trail,3 spanning subgraph,6 transitive, 24 spatial representation, 153 translatable fibre, 302 sphere representation, 330 transversal design, 269 spindle, 319 trap, 59 spindle surface, 73 tree,4 spine, 327 tree decomposition, 84 spring embedder, 160 tree-width, 4, 86 spring embedding algorithm, 66 triangle group, 203 stabilizer, 24, 201 triangulation, 14, 317 standard line, 251 tripod, 68 Steiner triple system, 246, 260, 269 Turán’s brickyard problem, 134, 315 Steinitz’s theorem, 11, 152 Tutte drawing, 159 st-ordering, 64 twisted type, 24 straightness of a ray, 292 two-fold triple system, 269 stratified graph, 59 two-way infinite path, 290 stratum, 59 type, 24, 174 strong component, 10 type of Cayley map, 211 strong homogeneous representation, 176 type of necklace,6 strong nodal domain, 157 strong representation, 153 strong symmetric genus, 29, 237 underlying topological space, 19 strongly cellular embedding, 20 Unendlichkeitslemma, 291 unimodal sequence, 55 strongly connected digraph, 10 union of graphs,6 strongly symmetric embedding, 227 unit-cost RAM, 65 strongly unimodal sequence, 55 unit-distance graph, 154, 332 subdivision of an edge,8 universal cover, 173 subexponential growth, 300 universally concentric Bilinski diagram, 307 subgraph,6 upper-embeddable graph, 39 support, 157 suppression of a vertex, 318 surface, 12 V -adjacent, 59 surgery, 26 valence, 19 symmetric embedding, 226 vertex, 1, 10, 164 symmetric Euler genus, 227 vertex addition algorithm, 65 symmetric map, 199 vertex fibre, 28 symmetric non-orientable genus, 227 vertex-accumulation point, 303 syntheme, 262 vertex-homogeneous map, 310 vertex-set,1 vertex-transitive graph,2 Tait colouring, 323 Vizing’s theorem,9 Index 347

voltage, 27 weight of walk, 331 voltage assignment, 182 well-quasi-order, 82 voltage graph, 26, 27 width of graph, 117 VSLI design, 63

Xuong tree, 36 Wagner’s conjecture, 82 Wagner’s theorem, 11 walk,3 weak homogeneous representation, 176 Y- ordering, 318 weak negative support, 157 Y -exchange, 72 weak nodal domain, 157 weak positive support, 157 weak realizability problem, 145 zip product of graphs, 142