GRAPHS AND THEIR COVERINGS Jin Ho Kwak Department of Mathematics, POSTECH, Pohang, 790{784 Korea [email protected] Roman Nedela Matej Bel University, Banska Bystrica, Slovakia Mathematical Institute, Slovak Academy of Sciences, Banska Bystrica, Slovakia [email protected] This work is supported by Com2MaC-KOSEF, Korea. Preface Combinatorial treatment of graph coverings had its primary incentive in the solution of Heawood's Map Colour Problem due to Ringel, Youngs and others [Ri74]. That coverings underlie the techniques that led to the eventual solution of the problem was recognized by Alpert and Gross [GA74]. These ideas further crystalized in 1974 in the work of Gross [Gr74] where voltage graphs were introduced as a means of a purely combinatorial description of regular graph coverings. In paral- lel, the very same idea appeared in Biggs' monograph [Bi74]. Much of the theory of combinatorial graph coverings in its own right was subse- quently developed by Gross and Tucker in the seventies. We refer the reader to [GT87, Wh84] and the references therein. The theory was extended to combinatorial graph bundles introduced by Pisanski and Vrabec [PV82] in the eighties. After two decades, since the book of Gross and Tucker [GT87] was published, it remains the most popular textbook as well as the reference book covering the topic. The authors of the presented monograph have several ambitions. Firstly, following the ideas in [MNS00] to establish and extend the combinatorial theory of graph coverings and voltage assignments onto a more general class of graphs which include edges with free ends (called semiedges). The new concept of a graph proved to be useful in applica- tions as well as in theoretical considerations. Some of the applications are shown in the book. From the theoretical point of view, graphs with semiedges arise as quotients of standard graphs by groups of automor- phisms which are semiregular on vertices and darts (arcs) but may ¯x edges. They may be viewed as 1-dimensional analogues of 2- and 3- dimensional orbifolds. The fruitful concept of an orbifold comes from studies of geometry of n-manifolds. Orbifolds and related concepts are implicitly included in the work of pioneers such as Henri Poincare, or Paul KÄobe. The ¯rst formal de¯nition of an orbifold-like object was i ii given by Ichiro Satake in 1956 in the context of Riemannian geome- try. William Thurston, later in the mid 1970s de¯ned and named the more general notion of orbifold as part of his study of hyperbolic struc- tures. It is not a surprise that the concept of graphs with semiedges was discovered in context of theoretical physics, as well (see [GK98] for instance). Secondly, following the ideas in [GS93] and in the subsequent papers [MNS00, Ma98, Sir01, Ma02, MMP04, MP06] we develop and organize a material on the combinatorial treatment of the lifting automorphism problem - a central problem considered in this book. The study of the automorphism lifting problem in the context of regular coverings of graphs had its main motivation in constructing in¯nite families of highly transitive graphs. The ¯rst notable contribution along these lines ap- peared, incidentally, in 1974 in Biggs' monograph [Bi74] and in a paper of Djokovi¶c[Dj74]. Whereas Biggs gave a combinatorial su±cient con- dition for a lifted group to be a split extension, Djokovi¶cfound a crite- rion, in terms of the fundamental group, for a group of automorphisms to lift at all. A decade later, several di®erent sources added further motivations for studying the lifting problem. These include: counting isomorphism classes of coverings and, more generally, graph bundles, as considered by Hofmeister [Ho91] and Kwak and Lee [KL90, KL92]; constructions of regular maps on surfaces based on covering space tech- niques due to Archdeacon, Gvozdjak, Nedela, Richter, Sir¶a·n,· Skoviera· and Surowski [ARSS94, AGJ97, GS93, NS96, NS97, Su88]; and con- struction of transitive graphs with prescribed degree of symmetry, for instance by Du, Malni·c,Nedela, Maru·si·c,Scapellato, Seifter, Tro¯- mov and Waller [DMW> 06, MM93, MMS> 06, MS93, ST97]. Lifting and/or projecting techniques play a prominent role also in the study of imprimitive graphs, cf. Gardiner and Praeger [GP95] among others. The lifting problem in terms of voltages in the context of general topo- logical spaces was considered by Malni·c[Ma98]. Venkatesh [Ve> 06] obtained structural results which re¯ne the work of Biggs [Bi74, Bi84] and Djokovi¶c[Dj74] on lifting groups. Recently, many original contribu- tions on the topic was published, both solving particular problems and developing pieces of theory. However, there is no monograph covering (more or less completely) fundamentals of the theory. The topic is omit- ted in the book by Gross and Tucker and in other related monographs iii in topological graph theory. So far, the most complete contribution to the theory was done in the thesis of A. Malni·c[Ma96], some parts are covered in the paper [MNS00]. The study of the lifting problem within the combinatorial framework was of course preceded by that in the topological setting. We refer the reader to the work of Armstrong, Birman, Hilden, MacBeath and Zieschang [Ar92, BH72, Mac61, Zi73] and many others. Finally, it is our ambition to shift by one dimension up, namely, to study 2-cell embeddings of graphs into surfaces. The corresponding 2- dimensional objects are called maps and they present a central object of topological graph theory. For the purpose of this future project, it was necessary ¯rst to rebuilt some classical parts of graph theory and the presented monograph can be used as preliminary material for investigation of maps and their symmetries. The book contains three chapters. First one introduce basic con- cepts such as graphs and groups acting on graphs. Some useful concepts and statements from permutation groups are briefly mentioned. The second chapter is central. There, a theory of graph coverings is devel- oped. It includes action of the fundamental group, classical approach to the theory of graph coverings and the associated theory of voltage spaces with some applications. The lifting automorphism problem is studied in detail, theory of voltage spaces us uni¯ed and generalized to graphs with semiedges. This way `a gap' in the familiar and popular Gross and Tucker's book `Topological graph theory' is ¯lled in. In the ¯nal chapter the lifting and covering techniques are used to approach some problems of classical graph theory. In particular, a material on flows on graphs and enumeration of graph coverings is included. The book can be used as a material for a course on graph coverings as well as it could be of interest of specialist interesting in the topic. We would like to take this opportunity to thank all people who give us valuable comments and suggestions for revising this book, including many students. Finally we are grateful to Korea Science and Engineering Founda- tion (KOSEF) in Korea for supporting us during preparation of this book. Jin Ho Kwak iv Roman Nedela E-mail: [email protected] [email protected] July 2005, in Pohang, Korea Contents Preface i 1 Graphs and Groups 1 1.1 Introduction . 1 1.2 Graphs and their fundamental groups . 2 1.3 Group actions . 7 1.4 Vertex-transitive graphs . 13 2 Coverings and liftings of automorphisms 23 2.1 Graph coverings and two group actions on the ¯bre . 23 2.2 Voltage spaces . 30 2.3 Covering isomorphism problem . 37 2.4 Lifting automorphism problem - classical approach . 42 2.5 Lifting of graph automorphisms in terms of voltages . 50 2.6 Lifting problem, case of abelian CT(p).......... 59 2.7 Lifting problem, case of elementary abelian CT(p).... 69 3 Applications 81 3.1 Enumeration of Graph Coverings . 81 3.2 Some applications of graph coverings . 90 3.2.1 Existence of regular graphs with large girth . 91 3.2.2 Large graphs of given degree and diameter 2 . 91 3.2.3 Spectrum of a graph and its covering graphs . 93 3.3 Flows . 95 Index 109 v Chapter 1 Graphs and Groups We begin with some basic de¯nitions and elementary properties of graphs and coverings. 1.1 Introduction In considerations in graph theory and more generally in combinatorics one has to deal with a problem of representation and controlling prop- erties of large structures, potentially based on unbounded number of elementary objects (vertices, points, edges). Several essentially di®erent techniques to deal with large objects were developed. Most usual are inductive constructions and arguments employing inductive arguments, counting arguments which led to the birth and development of the prob- abilistic method in graph theory and ¯nally, employing symmetries of structures one can use ¯nite groups to construct highly symmetrical graphs, designs, maps, geometries, etc, satisfying various combinato- rial properties. The latest approach is well-documented by a fruitful concept of a Cayley graph, introduced originally by group theorists in 19-th century to investigate groups. Topological graph theory deals with graphs as 1-dimensional topo- logical objects which are usually embedded to surfaces or to a more complex CW-complexes. The aim of this introductory chapter is to develop a consistent combinatorial theory of such graphs. In partic- ular, epimorphisms between graphs which are locally bijective, called graph coverings, will be investigated. We demonstrate the usefulness of 1 2 Chapter 1. Graphs and Groups graph coverings to solve several classical problems. Between the appli- cations the lifting automorphism problem is emphasized. There are two reasons: ¯rstly, the lifting automorphism problem and its applications in constructions of highly symmetrical graphs and maps received last decade a lot of attention in literature.