Group Actions on Graphs, Maps and Surfaces

Ministry of Education of Slovak Republic Scientiﬁc Board of P. J. Saf´arikUniversityˇ

Doc. RNDr. Roman Nedela, CSc. Group Actions on Graphs, Maps and Surfaces

Dissertation for the Degree of Doctor of Physical-Mathematical Sciences Branch: 11-11-9 Discrete Mathematics

Koˇsice,May 2005

Contents

1 Introduction 5

2 Half-arc-transitive actions of groups on graphs of valency four 7 2.1 Graphs and groups of automorphisms ...... 7 2.2 Half-arc-transitive action, G-orientation...... 7 2.3 Orbital graphs ...... 9 2.4 A construction of half-arc-transitive graphs of valency 4 . . . . . 10 2.5 Alternating cycles, classiﬁcation of tightly attached graphs . . . . 10 2.6 Regular maps and half-arc-transitive graphs of valency four . . . 11 2.7 P l and Al operators on graphs of valency 4 ...... 13 2.8 Graphs of valency 4 and girth 4 ...... 14 2.9 Classiﬁcation of point stabilizers ...... 15 2.10 Relations to group actions of other sorts ...... 17

3 Maps, Regular Maps and Hypermaps 21 3.1 Topological and combinatorial maps, permutation representation of maps ...... 21 3.2 Generalization to hypermaps, Walsh map of a hypermap . . . . . 28 3.3 Maps, hypermaps and groups ...... 30 3.4 Regular maps of large planar width and residual ﬁniteness of triangle groups ...... 37 3.5 Maps, hypermaps and Riemann surfaces ...... 40 3.6 Enumeration of maps of given genus ...... 43 3.7 Regular hypermaps on a ﬁxed surface ...... 48 3.8 Operations on maps and hypermaps, external symmetries of hypermaps ...... 51 3.9 Lifting automorphisms of maps ...... 54 3.10 Regular embeddings of graphs ...... 57

4 Minimal triangulations of given edge width 75

5 Publication record and citation index 79 5.1 Publication record of the author ...... 79 5.2 Citation index of the author ...... 83 5.3 Survey of publications and citations ...... 99 5.4 List of invited visits and selected talks in conferences ...... 99 5.5 Other activities ...... 101

3 R. Nedela: Group Actions on Surfaces

6 Appendix: Reprints of papers 103 On the point stabilizers of transitive groups with non-self-paired suborbits of length 2 (with D. Maruˇsiˇc) ...... 103 Exponents of orientable maps (with M. Skoviera)ˇ ...... 103 Classiﬁcation of regular maps of prime negative Euler characteristic (with A. Breda and J. Sir´aˇn)ˇ ...... 103 K-minimal triangulations of surfaces (with A. Malniˇc) ...... 103

4 Chapter 1

Introduction

The presented thesis deals with three topics: half-arc-transitive actions of groups on graphs, regular maps and hypermaps and triangulations of given planar edge-width. These topics are unified by the interaction of graphs, groups and surfaces. In the first two, the role of group theoretical methods and aspects is stressed, although we briefly mention other related mathematical objects such as Riemann surfaces or algebraic curves, for instance. Chapter 2 can be considered as a survey on half-arc-transitive group actions on 4-valent graphs, or equivalently, on transitive permutation groups with non- self-paired suborbits of length two. Most of the related work of the author was done in a fruitful collaboration with Dragan Maruˇsiˇc(Univ. Ljubljana) during years 1994-2000. The strongest result we have achieved is a classification of point stabilisers of such actions (see the attached reprint for details). Chapter 3 deals with maps and hypermaps posessing maximum number of symmetries, called regular maps and regular hypermaps. First, parts of the general theory of maps and hypermaps is built. Then an exhaustive survey follows. In each section one of the fundamental problems is discussed, the reader is provided by the most recent information on the current stage of art of the research in the field. Of course, any such a survey is subjective, some aspects are more stressed (namely the ones in which the author actively contributed), some others are suppressed. Regular maps and hypermaps were (and still are) in the center of my research activities. Most of my results in the field were done in collaboration with other mathematicians, to name just few of them I mention Antonio Breda, Shao-Fei Du, Jin-Ho Kwak, Gareth Jones, Aleksander Malniˇc,Alexander Mednykh, Martin Skovieraˇ and Jozef Siráˇn.ˇ Among many interesting results I have chosen (for the Appendix) a recent paper by Breda, Nedela and Siráˇnclassifyingˇ regular maps on surfaces of prime negative Euler characteristic. As a consequence, an infinite family of non-orientable surfaces admitting no regular maps was found. This way we have solved a problem of Conder and Everitt (1995). Another selected paper by Skovieraˇ and myself is devoted to a systematic study of exponents of maps (integer invariants related with certain map operations). Exponents of maps play an important role in classification of regular embeddings of graphs. Chapter 4 deals with minimal triangulations of given edge-width. In collaboration with Aleksander Malniˇcwe have proved (independently on Robertson- Seymour theory) that the class of triangulations on given surface and of bounded

5 R. Nedela: Group Actions on Surfaces edge-width has a ﬁnite basis with respect to the vertex-splitting operation. Our result generalises the well-known statement of Steinitz on polyhedral triangulations, see the attached paper for details. Chapter 5 is obligatory. It contains some information on research activities of the author. In the Appendix the above-mentioned papers of the author related with Chapters 2,3 and 4, respectively, are included. In the end I would like to thank to all my friends and collaborators giving me an opportunity to share a beauty of mathematics, supporting me during crises and coming with new ideas giving me an additional boost and inspiration to work on research projects. Between all my collaborators the one, who have mostly inﬂuenced my development as a mathematician has a special position, this is Martin Skoviera.ˇ Thanks Martin!

In Bansk´aBystrica, March 31, 2005

6 Chapter 2

Half-arc-transitive actions of groups on graphs of valency four

2.1 Graphs and groups of automorphisms

Throughout this section by a graph we mean an ordered pair (V,E), where V is a finite nonempty set and E is a symmetric irreflexive relation on V , whose transitive closure is the universal relation. Graphs are thus simple and connected, unless specified otherwise. By a directed graph X we mean an ordered pair (V,A), where V is a finite nonempty set and A, the set of arcs, is an asymmetric relation on V . A directed graph is balanced if for every vertex the number of incoming arcs is equal to the number of outgoing arcs. For a graph X, we let V (X), E(X), A(X) and Aut (X) denote the respective sets of vertices, edges and arcs, and the automorphism group of X. Given (undirected) graph X the set of arcs of X is said to be A = {(x, y), (y, x)|[x, y] ∈ E(X)}. An automorphism of a graph X is a permutation ψ of V such that [ψ(x), ψ(y)] ∈ E(X) for every edge [x, y] ∈ E(X). Similarly, an automorphism of an oriented graph X is a permutation ψ of V such that [ψ(x), ψ(y)] ∈ A(X) for every arc [x, y] ∈ A(X). A group of automorphisms G ≤ Aut (X) of a graph has an induced action on edges of X and on arcs of X. A graph X is said to be vertex- transitive, edge-transitive and arc-transitive, respectively, if its automorphism group Aut (X) acts vertex-transitively, edge-transitively and arc-transitively. Furthermore, all groups are assumed to be finite. For graph-theoretic and group-theoretic terms not defined here we refer the reader to [5, 12, 32, 33, 65].

2.2 Half-arc-transitive action, G-orientation.

An edge-transitive group G ≤ Aut (X) of automorphisms of a graph X is either vertex-transitive, or X is bipartite. Complete bipartite graphs Km,n with m 6= n are obvious representatives of bipartite edge- but not vertex-transitive graphs.

7 R. Nedela: Group Actions on Surfaces

If G is vertex- and edge-transitive on X then the induced action on the set of arcs has at most two orbits. Thus such actions split into two families: arc- transitive group actions acting with one orbit on A(X), and half-arc-transitive group actions giving two orbits on A(X). Half-arc-transitive group actions on graphs present the main objective of investigation in this chapter. Let G act half-transitively on X = (V,E) and let take an orbit O ⊂ A(X) of the action. It follows that O intersects each pair of oppositely directed arcs −→ (x, y), (y, x) in exactly one arc. Then X = (V,O) is an associated directed −→ graph. Note that X can be obtained from X = (V,O) by forgetting the ori- −→ entation arcs of X = (V,O). If O¯ denotes the complementary orbit to O then ←− −→ X = (V, O¯) can be obtained by reversing the orientation of each arc of X. −→ Sometimes we need to express that given directed graph X arises from a graph X and a half-arc-transitive group G ≤ Aut (X). In such a case we call any of −→ ←− X, X a G-orientation of X. There are graphs admitting two half-arc-transitive group actions such that the respective G-orientations are essentially diﬀerent (see Figure 2.1).

1 5 6 7 1 1 5 6 7 1

4 4 4 4

3 3 3 3

2 2 2 2

1 5 6 7 1 1 5 6 7 1

Figure 2.1: Two essentially diﬀerent G-orientations of C4 × C4.

Note that if X is half-arc-transitive then all groups G ≤ Aut (X) acting half-arc-transitively on X induce the same G-orientation of X. The following fundamental statement comes from Tutte [57, page 59].

Theorem 2.1 Let X = (V,E) be a graph and G ≤ Aut (X) acts half-arc- transitively. Then X is k-valent for some even integer k. −→ Proof. Take a G-orientation X of X. By its deﬁnition G acts transitively −→ on vertices and on arcs of X. Let v be a ﬁxed vertex and let m = indeg(v) and n = outdeg(v) be the numbers of incoming and outcoming arcs incident v, respectively. Since G is transitiveP on vertices, theseP numbers do not depend on the choice of v. Since m|V | = v∈V indeg(v) = v∈V outdeg(v) = n|V | we have m = n. Hence the valency of each vertex is k = 2m = 2n. ¤

Thus a G-orientation of a graph admitting a half-arc-transitive action of G is always balanced. Since connected graphs of valency 2 are just cycles the smallest non-trivial case to consider is valency four.

8 R. Nedela: Group Actions on Surfaces

Note that the above Theorem 2.1 does not generalize to infinite locally finite −→ −→ graphs. Given directed graph X by P l(X) we denote the partial line graph of −→ X with the vertex set A = A(X) and arc-set defined as follows: ((x, y), (z, w)) −→ is an arc of P l(X) if and only if y = z and x 6= w. If a group G acting half-arc- transitively is fixed then we apply the partial line graph operator on (undirected) graph X via its G-orientation. The resulting graph P l(X) is undirected. It arises −→ by forgetting the direction of arcs of P l(X).

Theorem 2.2 For every odd integer k ≥ 5 there is an inﬁnite half-arc-transitive graph X of valency k. If X is a cubic graph admitting a half-arc-transitive action of a group then X is a cubic tree. In particular, there is no cubic half- arc-transitive graph.

Proof. Let k = 2m + 1. Take a k-valent infinite tree T with k ≥ 5. One −→ can define inductively an unbalanced orientation of T thus forming T such that −→ indeg(v) = m and outdeg(v) = m + 1 for every vertex v in T . There is an associated group G ≤ Aut (T ) which induces the above orientation of T . Set −→ X = P l( T ) = P l(T ). Then G acts half-arc-transitively on X. Consider the subgraph Y of X induced by the vertices that come from 2m+1 arcs incident to a ∼ fixed vertex of T . By the definition of X, Y = Km,m+1 and the subgraphs of this sort corresponding to vertices of the original graph form blocks of imprimitivity in the action of Aut (X). In particular, X cannot be arc-transitive since Km,m+1 is not. Let us assume that k = 3 and X is cubic. Let G be a group acting half-arc- transitively on X. By vertex-transitivity indeg(v)(outdeg(v) is the same for −→ any vertex v of X. Without loss of generality we assume that indeg(v) = 1 and outdeg(v) = 2 for each vertex v. Let C be a cycle of X. Then G induces an orientation of arcs of C. The conditions indeg(v) = 1 and outdeg(v) = 2 force −→ −→ C to have transitive orientation, i. e. C ⊆ X is a directed cycle. Let x be an −→ −→ arc incident to a vertex in C but not belonging to C . By transitivity of the −→ action on arcs there is a directed cycle C0 passing through x. Hence C ∩ C0 is nonempty, a contradiction. Thus X is a cubic tree. ¤

2.3 Orbital graphs

Let G be a transitive permutation group acting on a set V and let v ∈ V . There is a 1-1 correspondence between the set of orbits of the stabilizer Gv on V , that is, the set of suborbits of G, and the set of orbitals of G, that is, the set of orbits in the natural action of G on V ×V , with the trivial suborbit {v} corresponding to the diagonal {(v, v): v ∈ V }. The paired orbital ∆t of an orbital ∆ is the orbital {(v, w):(w, v) ∈ ∆}. If ∆t = ∆ we say that ∆ is a self-paired orbital.A self-paired suborbit of G is a suborbit which corresponds to a self-paired orbital. The orbital graph X(G, V ; ∆) of (G, V ) relative to ∆, is the graph with vertex set V and arc set ∆. The suborbit ∆ is said to be connected if the underlying t undirected graph X0(G, V ; ∆) of X(G, V ; ∆) is connected. Of course, if ∆ = ∆ is a self-paired orbital then X(G, V ; ∆) can be viewed as an undirected graph which admits a vertex- and arc-transitive action of G. On the other hand, if t t ∆ 6= ∆ is a non-self-paired orbital then ∆ ∩ ∆ = ∅ and X0(G, V ; ∆) admits

9 R. Nedela: Group Actions on Surfaces

1 a 2 -arc-transitive action of G. Conversely, given an edge [u, v] of a graph X of 1 valency 2d admitting a 2 -arc-transitive action of some group G ≤ Aut (X), the two arcs (u, v) and (v, u) give rise (via the action of G) to two oriented graphs, namely the orbital graphs of G relative to two paired orbitals, where the length of the corresponding two suborbits is d. The above discussion is summarized in the following theorem.

Theorem 2.3 Let X = (V,E) be a 2d-valent graph and G ≤ Aut (X) acts half- arc-transitively on X. Then there exists a corresponding orbital ∆ in the action of G on V ×V with a non-self-paired suborbit of length d such that the respective ∼ orbital graph XO(G, V ; ∆) = X Vice-versa, each transitive action of a permutation group G with a non-self- paired connected suborbit of length d gives rise an orbital graph XO(G, V ; ∆) which admits a half-arc-transitive action of G.

Since a transitive action of a group G is equivalent to the action of G on left cosets of Gv by left multiplication a half-arc-transitive graph is determined by specifying G, H = Gv ≤ G and d elements a1, . . . , ad such that {(H, aiH)|i = 1, . . . d} is a non-self-paired suborbit of length d. In the speciﬁc instance of this situation, when an orbital ∆ has a non-self- paired connected suborbit of length 2 in the action of a group G on the set of left cosets of its subgroup H, we say that the associated 4-valent graph X0(G, H, ∆) 1 1 is (G, 2 ,H)-arc-transitive, or (G, 2 )-arc-transitive in short. Of course, when G = 1 Aut (X) then X is a 2 -arc-transitive graph. Hence to study half-arc-transitive actions of groups on graphs of valency four is the same as to study transitive permutation groups with a connected non-self-paired suborbit of length 2.

2.4 A construction of half-arc-transitive graphs of valency 4

∗ Let r ≥ 3 be an odd integer, t ≥ 3 be an integer and let s ∈ Zr satisfy i t j j j±s s ∈ {1, −1} modulo r. Set V = {vi ; i ∈ Zt, j ∈ Zr} and E = {vi vi+1 |i ∈ Zt, j ∈ Zr}. Denote X(s; t, r) = (V,E). For instance, X(2; 3, 9) is the smallest half-arc-transitive graph called the Holt’s graph, see [20], while X(2; 6, 9) is the smallest graph in Bouwer’s family [3]. The permutations ρ, σ and τ given j j+1 j sj j −j by the rules vi ρ = vi , vi σ = vi+1 and vi τ = vi generate a subgroup G = G(s, t, r) of automorphisms of X(s; t, r) acting half-arc-transitively on X(s; t, r). Note that hρ, σi acting transitively on vertices is metacyclic, so X(s; t, r) is a metacirculant. The half-arc-transitivity of graphs X(s; t, r) was considered in [1] and proved in two particular cases: t = 3 and r ≥ 9, or r prime and t a composite integer. Furthermore, in [49] is shown that X(s; 4, r) is half-arc- transitive provided s2 6= ±1.

2.5 Alternating cycles, classiﬁcation of tightly attached graphs

Let consider a 4-valent graph with a ﬁxed G ≤ Aut (X) acting half-arc-transitively on X. A walk W in an oriented graph is called alternating if every other

10 R. Nedela: Group Actions on Surfaces vertex is a tail and every other vertex is a head of two consecutive arcs in W . Since the G-orientation of X is balanced, each arc lies in a unique alternating cycle called a G-alternating cycle. The set of G-alternating cycles decompose the set of edges. They are of even length 2r, the parameter r ≥ 2 is called G-radius of X. The following result was proved by Maruˇsiˇc[32].

1 Theorem 2.4 (Maruˇsiˇc[32]) Let X be a (G, 2 )-transitive graph of valency 4 for some G ≤ Aut (X). Then there exists an integer r ≥ 2 such that all G-alternating cycles have length 2r for some r and the set of G- alternating cycles form a decomposition of the set of edges of X; either X has precisely two G-alternating cycles, both spanning V (X), ∼ which occurs if and only if X = Cay(Z2r; {1, −1, s, −s} for some odd ∗ 2 s ∈ Z2r \{1, −1} such that s = ±1; in this case X is arc-transitive; or X has at least three G-alternating cycles, which are all induced cycles. Maruˇsiˇchas observed in [32] that any two adjacent G-alternating intersect in the same number of vertices aG(X) and that this number divides the G-radius. If aG(X) = rG(X), aG(X) = 1 or aG(X) = 2 the graph is called, respectively, tightly G-attached, loosely G-attached or antipodally G-attached. Maruˇsiˇcand 1 Praeger [44] proved that a (G, 2 )-transitive graph of valency four is a cover of 1 (G, 2 )-transitive graph Y of valency 4 which is loosely, antipodally or tightly G-attached. Hence understanding the above three classes of half-arc transitive graphs of valency four is of crucial importance. A partial classiﬁcation in this direction is proved in [32].

Theorem 2.5 (Maruˇsiˇc[32]) Let X be a tightly attached half-transitive graph of valency four with an odd radius r. Then X =∼ X(s; t, r), where t ≥ 3 is an ∗ t integer, s ∈ Zr and s ∈ {1, −1}, and moreover none of the following three conditions is satisﬁed: s2 = ±1; (s; t, r) = (2, 3, 7), (2; 6, 7); (s; t, r) = (s; 6, 7k), where k ≥ 3, (7, k) = 1, s6 = 1 and the equation x2 + x − 2 = 0 has a unique solution q ∈ {s, −s, 1/s, −1/s} such that 7(q − 1) = 0 and q ≡ 5 (mod 7).

1 In [44] the classiﬁcation of (G, 2 )-transitive graphs of even radius is completed. Further results related on G-alternating and related invariants can be found in [40] and [44].

2.6 Regular maps and half-arc-transitive graphs of valency four

A map is a 2-cell decomposition of a surface. In this section we shall assume that the underlying surface of a map is orientable and that the degree of every vertex and size of every face is at least three. A map M is (orientably) regular if the (orientation preserving) map automorphism group acts arc-transitively on

11 R. Nedela: Group Actions on Surfaces

M. Given map M we can form a new map Med(M), called medial map of M, which vertices are centers of edges of M and two vertices are joined by an edge if the respective edges of M are consecutive in the boundary walk of a face of M. It is assumed that the edges of Med(M) are placed inside faces of M and that this is done in a way that edges of Med(M) do not cross. Since every edge lies in exactly four angles, the underlying graph X of Med(M) is 4-valent. The faces of Med(M) split into two disjoint classes: vertex-faces containing the vertices of the original map M and face-faces containing the centers of faces of M. Since the surface is orientable one can orient the arcs of the medial graph such that the boundary walks of vertex- and face-faces form transitive closed walks inducing the same, say counter-clockwise, orientation of vertex-faces and the reverse orientation of face-faces.

Figure 2.2: Cube and its medial

If M is regular with G = Aut (M) then the medial graph X of M is 1 (G, 2 ,Z2)-transitive of valency four. The G-orientation is just the one described 1 above. Vice-versa, given (G, 2 ,Z2)-transitive 4-valent graph X, there is a regular map M such that X is the medial graph of M and G = Aut (M). Hence 1 with any (G, 2 ,Z2)-transitive 4-valent graph there is an associated orientable surface. The above described relationship is studied in detail by Maruˇsiˇcand Nedela in [36]. A map is called chiral if it does not admit a reﬂection (an orientation reversing automorphism). The results proved in [36] are applied to show that apart from few small exceptions Coxeter chiral maps on torus of type {3, 6} (see [11, Chapter 7]) have half-arc-transitive medials. Note that every such a map M is (up to duality) determined by a couple of non-negative integers b, c; thus we write M = Mb,c = {3, 6}b,c or M = {6, 3}b,c in the notation of Coxeter-Moser [11]). The result follows. Theorem 2.6 A half-arc-transitive graph of valency 4 is toroidal if and only if it is isomorphic to the medial graph of the toroidal regular map Mb,c, where b, c satisfy bc(b − c) 6= 0 and b + c > 3.

1 The correspondence between regular maps and (G, 2 ,Z2)-transitive graphs of valency four is a particular instance of a more general relation between regular

12 R. Nedela: Group Actions on Surfaces

1 hypermaps and (G, 2 )-transitive graphs with cyclic G-stabilizers. This generalization is studied in [4].

2.7 P l and Al operators on graphs of valency 4

Given oriented graph X the line graph L(X) is a (oriented) graph whose vertices are arcs of X, and xy is an arc in L(X) if xy form a 2-arc in X. If X is balanced of valency four then L(X) is balanced of valency four as well. Moreover, the set of alternating 4-cycles decomposes the set arcs of X. Assume a balanced 4-valent oriented graph Y share the following two properties: (i) the set of its alternating 4-cycles decomposes the set of arcs of Y , (ii) two alternating 4-cycles meet in at most one vertex. Then there is a unique balanced oriented 4-valent graph X such that Y = L(X). It follows that there is a reverse operator Al deﬁned on balanced 4- valent oriented graphs satisfying (i) and (ii) and we can write X = Al(Y ) = Al(L(X)). In particular, the above deﬁned operators apply on 4-valent oriented graphs, with the orientation induced by a half-arc-transitive action of a group of automorphisms. 1 Given (G, 2 )-transitive graph of valency four the partial line graph P l(X) of X is the underlying undirected graph of a G-orientation of X. Maruˇsiˇc and Nedela [38] have studied the P l and Al operators in the context of half-arc- transitive graphs in detail. In particular, it is proved there that G acts on P l(X) as an orientation preserving group of automorphisms. The action is transitive on vertices. Since the number of vertices in P l(X) is doubled, the size of the vertex- stabilizer is halved. Using this observation one can easily deduce that the order of the stabilizer of G-action is a power of two, say 2h. The constant h is called 1 G-height of X. If h > 1 then P l(X) is (G, 2 )-transitive. For h = 1 the partial line graph of X is a Cayley graph Cay(G; {a, a−1, b, b−1}), where the generators G = ha, bi are non-involutory. Thus Cayley graphs Cay(G; {a, a−1, b, b−1}) can be considered to be the 4-valent graphs of height 0. It follows that every 4- valent half-arc-transitive graph with the automorphism group G of height h can be constructed from a 4-valent Cayley graph Cay(G; {a, a−1, b, b−1}) applying the Al operator h times. The following theorem summarizes the properties of 1 P l and Al operators on 4-valent (G, 2 )-transitive graphs. More results with a detailed explanation can be found in [38].

Theorem 2.7 Let X, Y be graphs of valency four. Then

1 (i) If X is (G, 2 ,H)-arc-transitive for some H ≤ G ≤ Aut (X) and |H| > 2, 1 then P l(X) is (G, 2 ,K)-arc-transitive with G-radius 2 for some K < H of 1 index 2 in H. Conversely, if Y is (G, 2 ,K)-arc-transitive with G-radius 2 and G-attachment number 1 for some K ≤ G ≤ Aut (Y ) such that 1 |K| ≥ 2, then X = Al(Y ) is (G, 2 ,H)-arc-transitive for some H ≤ G such that [H : K] = 2 and thus |H| > 2.

1 (ii) If X is (G, 2 ,Z2)-arc-transitive for some non-abelian subgroup G ≤ Aut (X), then there exist non-involutory generators generators a and b of G such that (ab−1)2 = 1 and P l(X) =∼ Cay(G; {a±1, b±1}). Conversely, if Y is 1 such a Cayley graph, then Al(Y ) is (G, 2 ,Z2)-transitive.

13 R. Nedela: Group Actions on Surfaces

Using Theorem 2.7 the following characterization of actions of small height is proved in [38].

1 Corollary 2.8 Let X be a (G, 2 )-transitive graph of height h ≤ 3. Then there are non-involutory generators a, b such that X =∼ Alh(Cay(G; a, b)) and a, b satisfy the following relations: (i) (ab−1)2 = 1 if h = 1, (ii) (ab−1)2 = 1 and (a2b−2)2 = 1 if h = 2, (iii) (ab−1)2 = 1, (a2b−2)2 = 1 and (a3b−3)2 = 1 or (ab−1)2 = 1, (a2b−2)2 = 1 and a3b−3a3b−1a−1b−1 = 1 if h = 3.

Remark. To stress the orientation of edges used to apply the Al operator we write Alh(Cay(G; a, b) for the respective undirected graph.

Example 2.9 Corollary 2.8(iii) is used by Conder and Maruˇsiˇcin [10] to construct a ﬁrst example of a half-arc-transitive graph of valency 4 with a non- abelian stabilizer. The group G = ha, bi is a subgroup of the symmetric group S32 given by a = (1, 2, 3, 4, 5, 6, 7, 8)(9, 10, 11, 12, 13, 14, 15, 16)(17, 18, 19, 20, 21, 22, 23, 24) (25, 26, 27, 28, 29, 30, 31, 32), and b = (1, 2, 11, 18, 21, 28, 27, 22, 5, 14, 15, 16, 9, 10, 3, 26, 29, 20, 19, 30, 13, 6, 7, 8) (4, 23, 32, 17, 12, 31, 24, 25). The proof that there are no more graph automorphisms is computer assisted.

2.8 Graphs of valency 4 and girth 4

In the previous section we have seen that 4-valent graphs of girth 4 form an important subset of 4-valent graphs admitting half-arc-transitive group action of a group G. An important question arises: Under what condition in such a graph all the 4-cycles are necessarily G-alternating? A 4-cycle C is called G-directed if the action of G induces a transitive orientation of C. ∗ Exceptional graphs. Let r, t be integers, r odd and s ∈ Zr satisfying st ∈ {1, −1}. Set X(t, r) = X(1; t, r). For r and t even let Y (t, r) denote j j j the graph with vertex set {vi : i ∈ Zt, j ∈ Zr} and edges of the form vi vi+1, j j+(−1)i+1 vi vi+1 , i ∈ Zt, j ∈ Zr. Moreover, let Z(t; r) denote the graph with the j j j j j+(−1)i+1 vertex set {vi : i ∈ Zt, j ∈ Zr} and edges of the form vi vi+1, vi vi+1 for j j+r/2 j j+1+r/2 i ∈ Zt \ {−1}, j ∈ Zr and v−1v0 , v−1v0 , j ∈ Zr. Finally, denote by Y = C2r ⊗ C2r, r ≥ 3 the halved quotient of the cartesian product C2r × C2r arising by identifying the (antipodal) couples of vertices (i, j), (i+r, j +r) for all i, j ∈ Z2r. Let us note that alternatively Y can be constructed as a medial graph of the quadrangular regular embedding of X = Cr ×Cr into the torus. While X is the underlying graph of the Coxeter map {4, 4}r,0, Y is the underlying graph of {4, 4}r,r. We say that a graph of valency 4 admitting a half-arc-transitive group action is exceptional if it belongs to one of the following families of (arc-transitive) graphs.

14 R. Nedela: Group Actions on Surfaces

∗ (i) the family F1 of circulants Cay(Z2r; {±1; ±s}),where s ∈ Z2r satisﬁes s2 ∈ {1, −1} ;

(ii) the family F2 of graphs X(t; r),where t ≥ 3 is an integer and r ≥ 3 is an odd integer;

(iii) the family F3 of graphs Y (t; r) ,where t ≥ 4 and r ≥ 4 are even integers;

(iv) the family F4 of graphs Z(t; r) ,where t ≥ 4 and r ≥ 4 are even integers;

(v) the family F5 of lexicographic products Ct[2K1],where t ≥ 3 is an integer;

(vi) the family F6 of Cartesian products C2r × C2r, r ≥ 2;

(vii) the family F7 of graphs C2r ⊗ C2r , r ≥ 3. The following statement proved in [39] gives an answer to the above problem.

1 Theorem 2.10 Let X be a (G, 2 )-transitive graph of valency 4 and girth 4, where G ≤ Aut (X). Then either every 4-cycle of X is G-alternating or every 4- cycle of X is G-directed or X is one of the exceptional graphs above. Moreover, if every 4-cycle of X is G-directed then either the G-height is 1 or X is exceptional.

Corollary 2.11 ([39]) Let X be a half-arc-transitive graph of valency 4 and girth 4. Then either (i) every 4-cycle is alternating or (ii) every 4-cycle is directed. Moreover, in case (ii) X is of height 1.

2.9 Classiﬁcation of point stabilizers

Information on the structure of point stabilizers of transitive permutation groups is important in classification results. In this section we first briefly mention known results then we shall concentrate to vertex stabilisers of half-arc-transitive group actions on 4-valent connected graphs. Their structure was described by Maruˇsiˇcand Nedela in [37], see the attached paper. Arc-transitive actions. A classical result of Tutte [55] establishes a bound 3 · 24 on the size of a vertex stabilizer of an arc-transitive group acting on a connected cubic graph. Later, it turned out (see Sims [50, 51] and Wong [66]) that exactly five groups, namely Z3, S3, D12, S4 and S4 × Z2, can serve as point stabilizers of such actions. For valency > 3 the size of vertex stabilizers is unbounded. However, when we are restricted to primitive group actions there are many particular results [22, 23, 45, 48, 51, 61] bounding the size of a stabilizer by a function of valency. Hence, there is an evidence that the following Sims’ conjecture could be true: Sims’ conjecture. Vertex stabilizers of primitive groups acting arc-transitively on graphs of valency k are bounded by a function of k. Sims’ conjecture was finally verified by Cameron, Praeger, Seitz and Saxl [6] using the classification of simple non-abelian groups. A bound in terms of

15 R. Nedela: Group Actions on Surfaces

valency in case of arbitrary transitive permutation groups and doubly primitive vertex stabilizer is established in [16, 17]. On the other hand, Weiss [63] conjectured that, if the vertex-stabilizer Gv acts primitively on the neighborhood of v in an arc-transitive action of G on a connected graph X then its size is bounded by a function of the valency of X, see the survey [47] for details. Recently, a reduction of the problem to simple groups is done in [9]. Since the Sims’ conjecture was verified a natural problem of determining possible vertex-stabilizers of primitive group actions on graphs of fixed (small) valency arises. As was already mentioned this problem was solved for valency 3. As concerns valency four, the upper bound 2436 was published in [23]. In the recent paper of Li, Lu and Maruˇsiˇc[27] the precise list of 10 possible groups which appear as vertex-stabilizers of primitive arc-transitive group actions on 4-valent graphs is derived. It follows that a sharp upper bound for the stabilizer 4 2 is 2 3 , achieved by the group S4 × S3. Half-arc-transitive actions on 4-valent graphs. Generally, the structure of vertex-stabilizers of half-arc-transitive group actions on connected graphs is not known. A half-arc-transitive action on connected graphs of valency four is not primitive and the size of the respective vertex-stabilizer is not bounded. As was already mentioned the vertex-stabilizer is a 2-group. Natural candidates are elementary abelian 2-groups, and indeed, it is easy to see that the lexicographic product Ch+1[K¯2] admits a half-arc-transitive action of G-height h. To con- 1 struct, (G, 2 ,H)-transitive 4-valent graphs with non-abelian stabilizers is less trivial. Actually, as a consequence of the following classification result proved by Maruˇsiˇcand Nedela [37] we obtain that the stabilizer H is almost abelian meaning the nilpotency class of H is at most two.

Theorem 2.12 Let G act half-arc-transitively on a connected 4-valent graph 2 with a vertex stabilizer H. Then there are positive integers h, d with 3 h ≤ d ≤ h and a set of generators hτ1, τ2, . . . , τhi of H satisfying the following relations:

2 (R1) τi = 1 for i = 1, 2, . . . , h, 2 (R2) (τiτj) = 1 if 0 < |i − j| < d,

2 ²(j−i,0) ²(j−i,1) ²(j−i,2d−2h+j−1) (R3) (τiτj) = τh−d+i τh−d+i+1 . . . τ(h−d+i)+2d−2h+j−1

for all 1 ≤ i < j ≤ h,where j − i ≥ d and ²(r, s) ∈ {0, 1}. Furthermore, G = ha, Hi = ha, bi, for some a ∈ G \ H, b = τ1a and conju- gation by a sends τi onto τi+1 for i = 1, 2, . . . , h − 1. On the other hand, let (G, H) be a pair of abstract groups satisfying the above conditions. Then the action of G on left cosets of H is faithful with a connected non-self-paired suborbit of length two and point stabilizer H, except ∼ h ∼ when H = Z2 is normal in G, or when h = 1, H = Z2 and G is dihedral. Theorem 2.13 Every group H with presentation of the form

H = hτ1, τ2, . . . , τh|(R1), (R2), (R3)i

embeds into the vertex stabilizer Aut (X)v of a 4-valent vertex-transitive graph X, hence H =∼ H¯ ≤ Aut (X), and there is a ∈ Aut (X) \ H¯ such that X is 1 ¯ (G, 2 ,H)-arc-transitive for G = ha, Hi.

16 R. Nedela: Group Actions on Surfaces

Note that the above automorphisms have the following meaning: given G- orientation a, b takes a ﬁxed arc x headed at a vertex v , respectively, onto the two arcs xa, xb outgoing from v. The element ai takes a ﬁxed (h − 1)-arc x onto ai y = x . The stabilizer of y is of order two and the involution τi is the unique non-trivial element in the stabilizer of y. Half-arc-transitive graphs with prescribed stabilizers. Most of the known half-arc-transitive graphs are of height 1. In [30] 4-valent half-arc- transitive graphs with stabilizers Z2 × Z2 are constructed. Recently, Maruˇsiˇc gives [34] a construction of 4-valent half-arc-transitive graphs with arbitrarily large elementary abelian stabilizers. First example of a 4-valent half-arc- transitive graph with a non-abelian stabilizer (isomorphic to the dihedral group D8) was constructed by Conder and Maruˇsiˇc[10], see Example 2.9. The following problem remains open.

Problem 2.1 Given group H with presentation of the form

H = hτ1, τ2, . . . , τh|(R1), (R2), (R3)i is there a 4-valent half-arc-transitive with vertex-stabilizer isomorphic to H?

2.10 Relations to group actions of other sorts

The same group admits, in general, many diﬀerent actions. In the following few lines we present a brief (and incomplete) list of objects related to half-arc- 1 transitive group actions. As already mentioned if X is a (G, 2 )-arc-transitive 4-valent graph oh height h then P lh(X) is a Cayley graph of valency 4 whose generators satisfy the relation (ab−1)2 = 1. Hence Cayley graphs of valency 1 4 based on a group G and (G, 2 )-arc-transitive graphs are closely related, for 1 more details see for instance [14, 25, 26, 52]. Given (G, 2 ,H)-transitive graph X with H being cyclic, the construction of medial (hyper)map gives an associated graph Y admitting a one-regular action of G with a cyclic stabiliser [36, 4]. A relationship of half-arc-transitive group actions to semi-symmetric group actions on bipartite graphs is investigated in [42, 43]. The latter graphs can be interpreted as incidence graphs of some geometries. Half-arc-transitive group actions arise naturally in an investigation of a homogeneous factorisations of index 2 which are not symmetric and for which each part is an orbit given by an action of some group [18]. Finally, half-arc-transitive group actions in the frame of association schemes and orbital graphs are studied in [19].

17 Bibliography

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[2] B. Alspach, D. Maruˇsiˇc,L. Nowitz, Constructing graphs which are 1/2-transitive. J. Austral. Math. Soc. Ser. A 56 3 (1994), 391–402.

[3] Bouwer, I. Z. Vertex and edge transitive, but not 1-transitive, graphs. Canad. Math. Bull. 13 (1970), 231–237.

[4] A. Breda, R. Nedela, Half-arc-transitive graphs and regular hypermaps, European J. Combin. 25 (2004), 423–436.

[5] N. Biggs and A.T. White, “Permutation groups and combinatorial structures”, Cam- bridge Univ. Press, 1979.

[6] P.J. Cameron, C. E. Praeger, G. M. Seitz, J. Saxl, On the Sim’S conjecture and distance transitive graphs, Bull London Math. Soc. 15 (1983), 499–506.

[7] M. D. E. Conder and C. G. Walker, Vertex-transitive non-Cayley graphs with arbitrarily large vertex stabilizers, J. Algebraic Combin. 8 (1998), 29–38.

[8] M.D.E. Conder, C.E. Praeger, Remarks on path-transitivity in ﬁnite graphs. European J. Combin. 17 (1996), 371–378.

[9] M.D.E. Conder, C.H. Li, Ch.E. Praeger, On the Weiss conjecture for ﬁnite locally primitive graphs. Proc. Edinburgh Math. Soc. (2) 43 (2000), 129–138.

[10] M.D.E. Conder and D. Maruˇsiˇc, A tetravalent half-arc-transitive graph with non- abelian vertex stabilizer, J. Combin. Theory B 88 (2003), 67–76.

[11] H. S. M. Coxeter and W. O. J. Moser, “Generators and Relations for Discrete Groups” (Fourth Edition), Springer-Verlag, Berlin, 1984.

[12] J.D. Dixon and B. Mortimer, “Permutation groups”, Springer-Verlag, New York, 1996.

[13] S.F. Du, M.Y. Xu, Vertex-Primitive 1/2-arc-transitive graphs of smallest order, Com- mun. Algebra 27 (1999), 163–171.

[14] X.G. Fang, C.H.Li, M.Y.Xu, On edge-transitive Cayley graphs of valency four, Euro- pean J. Combin. 25 (2004), 1107–1116.

[15] Y.Q. Feng, J.H. Kwak, Constructing an Inﬁnite Family of Cubic 1-Regular Graphs, European J. Combin. 23 (2002), 559–565.

[16] A. Gardiner, Doubly primitive vertex stabilisers in graphs, Math. Zeitschr. 135 (1974), 257–266.

[17] A. Gardiner, Arc-transitivity in graphs, II, Quart. J. Math. Oxford Ser. 25 (1974), 163–167.

[18] M. Giudici, C.H. Li, P. Potoˇcnikand Ch.E. Praeger, Homogeneous factorisations of graphs and digraphs, European J. Combin., In Press.

18 R. Nedela: Group Actions on Surfaces

[19] M. Hirasaka, M. Muzychuk, Association schemes generated by a non-symmetric relation of valency 2, Discrete Math. 244 (2002), 109–135.

[20] D.F. Holt, A graph which is edge transitive but not arc transitive. J. Graph Theory 5 (2) (1981), 201–204.

[21] D. Holton, Research problem 9, Discrete Math. 38 (1982), 125.

[22] W. Knapp, On the point stabiliser in a primitive permutation group, Math. Zeitschr. 133 (1973), 137–168.

[23] W. Knapp, Primitive Permutationsgruppen mit einem zweifach primitiven Subkon- stituenten, J. Algebra 38 (1976), 146–162.

[24] C.H. Li, On isomorphisms of ﬁnite Cayley graphs - a survey, Discrete Math. 256 (1-2) (2002), 301–334.

[25] C.H. Li CH, H. S. Sim, On half-transitive metacirculant graphs of prime-power order, J. Combin. Theory B 81 (2001), 45–57.

[26] C.H. Li, H.S. Sim, Automorphisms of Cayley graphs of metacyclic groups of prime- power order, J. Australian Math. Soc. A 71 (2001), 223–231.

[27] C.H. Li, Z.P. Lu, D. Maruˇsiˇc,On primitive permutation groups with small suborbits and their orbital graphs, Journal of Algebra 279 (2004), 749–770.

[28] M.W. Liebeck, C.E. Praeger, J. Saxl, Primitive permutation groups with a common suborbit, and edge-transitive graphs, Proc. London Math. Soc. 84 (2002), 405–438.

[29] A. Malniˇc,D. Maruˇsiˇc,Constructing 4-valent (1)/(2)-transitive graphs with a nonsolvable automorphism group, J. Combin. Theory B 75 (1) (1999), 46–55.

[30] A. Malniˇcand D. Maruˇsiˇc,Constructing 1/2-transitive graphs of valency 4 and vertex stabilizer Z2 × Z2, Discrete Math. 245 (2002), 203–216.

[31] D. Maruˇsiˇc,M.Y. Xu, 1/2-transitive graph of valency 4 with a nonsolvable group of automorphisms, J. Graph Theory 25 (2) (1997), 133–138.

[32] D. Maruˇsiˇc,Half-transitive group actions on ﬁnite graphs of valency 4, J. Combin. Theory B 73 (1998), 41–76.

[33] D. Maruˇsiˇc,Recent developments in half-transitive graphs, Discrete Math. 182 (1998), 219–231.

[34] D. Maruˇsiˇc,Half-arc-transitive graphs of valency 4 with large vertex stabilizers, Sub- mitted.

[35] D. Maruˇsiˇc,Constructing cubic edge- but not vertex-transitive graphs, J. Graph Theory 35 (2) (2000), 152–160.

[36] D. Maruˇsiˇc,R. Nedela, Maps and half-transitive graphs of valency 4, European J. Combin. 19 (3) (1998), 345–354.

[37] D. Maruˇsiˇcand R. Nedela, On the point stabilizers of transitive groups with non-self- paired suborbits of length 2, J. Group Theory 4 (1) (2001), 19–43.

[38] D. Maruˇsiˇcand R. Nedela, Partial line graph operator and half-arc-transitive group actions, Math. Slovaca 51 (2001), 241–251.

[39] D. Maruˇsiˇc,R. Nedela, Finite graphs of valency 4 and girth 4 admitting half-transitive group actions, J. Australian Math. Soc. 73 (2002), 155–170.

[40] D. Maruˇsiˇc,A.O. Waller, Half-transitive graphs of valency 4 with prescribed attachment numbers, J. Graph Theory 34 (1) (2000), 89–99.

[41] D. Maruˇsiˇc,T. Pisanski, Weakly ﬂag-transitive conﬁgurations and half-arc-transitive graphs, European J. Combin. 20 (6) (1999), 459–470.

19 R. Nedela: Group Actions on Surfaces

[42] D. Maruˇsiˇc,T. Pisanski, Symmetries of hexagonal molecular graphs on the torus, Croat. Chem. Acta 73 (4) (2000), 969–981. [43] D. Maruˇsiˇc,P. Potoˇcnik,Bridging semisymmetric and half-arc-transitive actions on graphs, European J. Combin. 23 (6) (2002), 719–732. [44] D. Maruˇsiˇc,C.E. Praeger, Tetravalent graphs admitting half-transitive group actions: Alternating cycles, J. Combin. Theory B 75 (2) (1999), 188–205. [45] P. M. Neumann, Finite permutation groups, edge-coloured graphs and matrices, “Top- ics in Group Theory and Computation”, (Proc. Summer School, University Coll., Gal- way, 1973), 82–118, Academic Press, London, 1977. [46] C. E. Praeger and M. Y. Xu, A characterisation of a class of symmetric graphs of twice prime valency, European J. Combin. 10 (1989), 91–102. [47] C. E. Praeger, Finite permutation groups - a survey, Groups - Canberra 1989, Lecture Notes in Math. 1456, Springer-Verlag, Berlin, 1990. [48] W. L. Quirin, Extension of some results of Manning and Wielandt on primitive permutation groups, Math. Zeitschr. 123 (1971), 223–230. [49] M. Sajna,ˇ Half-transitivity of some metacirculants, Discrete Math. 185 (1998), 117– 136. [50] C. C. Sims, Graphs and finite permutation groups, Math. Zeitschr. 95 (1967), 76–86. [51] C. C. Sims, Graphs and finite permutation groups, II, Math. Zeitschr. 103 (1968), 276–281. [52] N.D. Tan, On non-Cayley tetravalent metacirculant graphs, Graph Combinator. 18 (4) (2002), 795–802. [53] D. E. Taylor and M. Y. Xu, Vertex-primitive 1/2-transitive graphs, J. Austral. Math. Soc. Ser.A 57 (1994), 113–124. [54] C. Thomassen, M.E. Watkins, Infinite vertex-transitive, edge-transitive non-1-transitive graphs. Proc. Amer. Math. Soc. 105 (1989), 258–261. [55] W. T. Tutte, A family of cubical graphs, Proc. Camb. Phil. Soc. 43 (1947), 459–474. [56] W. T. Tutte, On the symmetry of cubic graphs, Canad. J. Math. 11 (1959), 621–624. [57] W.T. Tutte, “Connectivity in graphs”, Toronto University Press, 1966. [58] M.Y. Xu, Half-transitive graphs of prime-cube order. J. Algebraic Combin. 1 (1992), 275–282. [59] M.Y. Xu, Some new results on 1/2-transitive graphs. Adv. in Math. (China) 23 (6) (1994), 505–516. [60] M.Y. Xu, Some work on vertex-transitive graphs by Chinese mathematicians. “Group theory in China”, 224–254, Math. Appl. (China Ser.), 365, Kluwer Acad. Publ., Dor- drecht, 1996. [61] R. J. Wang, The primitive permutation groups with an orbital of length 4, Comm. Algebra 20 (1992), 889–921. [62] R. J. Wang, Half-transitive graphs of order a product of two distinct primes, Comm. Algebra 22 (1994), 915–927. [63] R. Weiss, The non-existence of 8-transitive graphs, Combinatorica 1 (1981), 309–311. [64] R. Weiss, s-transitive graphs, “Algebraic methods in graph theory”, Colloq. Soc. J. Bolyai 25 (1981), 827–847. [65] H. Wielandt, “Finite permutation groups”, Academic Press, New York, 1964.

[66] W. J. Wong, Determination of a class of primitive permutation groups, Math. Zeitschr. 99 (1967), 235–246.

20 Chapter 3

Maps, Regular Maps and Hypermaps

3.1 Topological and combinatorial maps, permutation representation of maps Topological maps A map on a surface is a cellular decomposition of a closed surface into 0- cells called vertices, 1-cells called edges and 2-cells called faces. The vertices and edges of a map form its underlying graph. A map is said to be orientable if the supporting surface is orientable, and is oriented if one of two possible orientations of the surface has been specified; otherwise, a map is nonoriented. Typically, a map on a surface is constructed by embedding of a connected graph in the surface. Graphs considered in this chapter are finite, non-trivial and connected unless the opposite follows from the immediate context. Edges of our graphs are of three kinds: links, loops and semiedges. Multiple adjacencies are allowed. A link is incident with two vertices while a loop or a semiedge is incident with a single vertex. A link or a loop gives rise to two oppositely directed darts that are reverse to each other. A semiedge incident with a vertex u gives rise to a single dart initiating at u that is reverse to itself. From the topological point of view, a semiedge is identical with a pendant edge except that its pendant end-point is not listed as a vertex. Summing up, a graph seen as a topological space is just a finite 1-dimensional cell complex. The following result relating numerical invariants of maps with the Euler characteristic of the supporting surface is well-known.

Theorem 3.1 (Euler formula) Let M be a map on a closed surface S of genus g with v vertices, e edges, s ≤ e semiedges and f faces. Then (1) v − e + s + f = 2 − 2g, if S is orientable;

(2) v − e + s + f = 2 − g, if S is nonorientable.

21 R. Nedela: Group Actions on Surfaces

Graphs For the sake of technical convenience we shall usually replace topological graphs and maps by their combinatorial counterparts. Formally, a (combinatorial) graph is a quadruple G = (D,V ; I,L) where D = D(G) and V = V (G) are disjoint non-empty finite sets, I : D → V is a surjective mapping, and L = LG is an involutory permutation on D. The elements of D and V are darts and vertices, respectively, I is the incidence function assigning to every dart its initial vertex, and L is the dart-reversing involution; the orbits of the group hLi on D are edges of G. If a dart x is a fixed point of L, that is, L(x) = x, then the corresponding edge is a semiedge. If IL(x) = I(x) but L(x) 6= x, then the edge is a loop. The remaining edges are links. Let us remark that a similar definition of graphs appears in Jones and Singerman [91] (see also [150, 138]). The usual graph-theoretical concepts such as cycles, connectedness, etc., easily translate to our present formalism.

Category of Maps and Ormaps As far as maps on surfaces are concerned, there are two essentially different approaches to their combinatorial description. The first approach, based on a rotation-involution pair acting on darts, involves the orientation of the supporting surface and so is suitable only for maps on orientable surfaces [70, 91]. The corresponding combinatorial structure is called a combinatorial (or, sometimes, algebraic) oriented map. The other approach, using three involutions acting on mutually incident (vertex, edge, face)-triples called flags, is orientation insensi- tive and thus allows us to represent maps on non-orientable surfaces as well [92]. The resulting combinatorial structure will be called a combinatorial nonoriented map. Accordingly, we shall usually employ the same notation for a topological map and for the corresponding combinatorial structure on it. We start with necessary definitions concerning oriented maps. By a (combinatorial) oriented map we henceforth mean a triple (D; R,L) where D = D(M) is a non-empty finite set of darts, and R and L are two permutations of D such that L is an involution and the group Mon(M) = hR,Li acts transitively on D. The group Mon(M) is called the oriented monodromy group of M. The permutation R is called the rotation of M. The orbits of the group hRi are the vertices of M, and elements of an orbit v of hRi are the darts radiating (or emanating) from v, that is, v is their initial vertex. The cycle of R permuting the darts emanating from v is the local rotation Rv at v. The permutation L is the dart-reversing involution of M, and the orbits of hLi are the edges of M. The orbits of hRLi define the face-boundaries of M. The incidence between vertices, edges and faces is given by nontrivial set intersection. The vertices, darts and the incidence function define the underlying graph M, which is always connected due to the transitive action of the monodromy group. An oriented map can be equivalently described as a pair (G; R) where G = (D,V ; I,L) is a connected graph and R is a permutation of the dart-set of G cyclically permuting darts with the same initial vertex, that is, IR(x) = I(x) for every dart x of G. Combinatorial nonoriented maps are built from three involutions acting on a non-empty finite set F of flags [92]. A (combinatorial) nonoriented map is a quadruple (F ; λ, ρ, τ) where λ, ρ and τ are fixed-point free involutory per-

22 R. Nedela: Group Actions on Surfaces

Figure 3.1: The ﬁve Platonic solids mutations of F = F (M) called the longitudinal, the rotary and the transversal involution, respectively, which satisfy the following conditions:

(i) λτ = τλ; and

(ii) the group hλ, ρ, τi acts transitively on F . This group is the nonoriented monodromy group Mon(M) of M. We deﬁne the vertices of M to be the orbits of the subgroup hρ, τi, the edges of M to be the orbits of hλ, τi, and the face-boundaries to be the orbits of hρ, λi under the action on F , the incidence being given by nontrivial set intersection. Note that each orbit z of hλ, τi has cardinality 2 or 4 according to whether z is a semiedge or not.

Example 3.2 The underlying graph of the tetrahedron is the complete graph K4 = (D,V ; I,L) on 4 vertices. We may set V = {1, 2, 3, 4}, D = {12, 21, 13, 31, 14, 41, 23, 32, 24, 42, 34, 43}, L(ij) = ji and I(ij) = i, for any ij ∈ D. Then the rotation at vertices compatible with the counterclockwise global orientation in

23 R. Nedela: Group Actions on Surfaces

Figure 3.2 is R = (12, 13, 14)(23, 21, 24)(31, 32, 34)(41, 43, 42). Vice-versa, having R we can identify the triangular faces of the map (D; R,L) via cycles of the permutation RL = (12, 24, 41)(21, 13, 32)(31, 14, 43)(23, 34, 42). Figure 3.3 shows the same topological map (the tetrahedron) described by means of 3 involutions acting on 24 ﬂags as a map (F ; λ, ρ, τ) in the category of Maps.

23 21 24

12 41 4 43 32

14 34

1 13 31 3

Figure 3.2: The tetrahedron described as an oriented map (D; R,L)

τ λ ρ

Figure 3.3: The tetrahedron described as a map (F ; λ, ρ, τ)

The meaning of the condition on λ, ρ, τ requiring these involutions to be ﬁxed-points free becomes clear, if one decides to extend our theory onto maps on surfaces with boundary. This was done by Bryant and Singerman in [35]. To do the generalisation we have to allow ﬁxed points of λ,ρ,τ. The underlying surface of a map has a non-empty boundary if and only if at least one of λ,ρ,τ

24 R. Nedela: Group Actions on Surfaces

fixes a flag. In fact, the category of nonoriented maps is not complete if we do not consider maps on surfaces with boundary, since a homomorphic image of a map on a closed surface can be a map on a surface with a non-empty boundary. As an example, consider an embedding of a cycle in the sphere and its quotient on the disk given by the reflection interchanging the two faces of the map and fixing the embedded graph point-wise. Clearly, the even-word subgroup hρτ, τλi of Mon(M) has always index at most two. If the index is two, then M is said to be orientable.

A B

E C

D D

C E

B A

Figure 3.4: K6 in the Projective plane

With every oriented map (D; R,L) we associate the corresponding nonoriented map M\ = (F \; λ\, ρ\, τ \) by setting F \ = D × {1, −1} and deﬁning for a ﬂag (x, j) ∈ D × {1, −1}:

λ\(x, j) = (L(x), −j), ρ\(x, j) = (Rj(x), −j), and τ \(x, j) = (x, −j).

Conversely, from an orientable nonoriented map M = (F ; λ, ρ, τ) we can construct a pair of oriented maps M0 = (D; R,L) and M00 = (D; R−1,L) that are the mirror image of each other. We take D to be the set F/τ of orbits of τ on F . Let us denote by F + ⊂ F one of the two orbits induced by the action of the even-word subgroup of Mon(M). For a dart {z, τ(z)} = [z], where z ∈ F +, we set R([z]) = [ρτ(z)] and L([z]) = [λτ(z)]. Instead of R we could have taken the rotation R0([z]) = [τρ(z)], but since R0 = R−1 we get nothing but the mirror image – as expected.

Test of orientability Let M = (F ; λ, ρ, τ) be an nonoriented map. We want to determine whether the respective supporting surface S is orientable, or not. The following simple algorithm is well-known, see [70, 146]. Consider the associated 3-valent graph G, whose set of vertices is F , each vertex f ∈ F is incident with darts (f, λ), (f, ρ) and (f, τ) and the dart reversing involution takes (f, λ) 7→ (λf, λ), (f, ρ) 7→

25 R. Nedela: Group Actions on Surfaces

Figure 3.5: Regular embedding of K5 in the torus appears in two ‘enantiomers’, in the category of ORMAPS they are not isomorphic.

(ρf, ρ), (f, τ) 7→ (τf, τ). Note that G is nothing but the dual of the barycentric subdivision of M. The result follows.

Proposition 3.3 An nonoriented map M = (F ; λ, ρ, τ) is orientable if and only if the associated 3-valent graph G is bipartite.

Homomorphisms of maps

Let M1 = (D1; R1,L1) and M2 = (D2; R2,L2) be two oriented maps. A homomorphism ϕ : M1 → M2 of oriented maps is a mapping ϕ : D1 → D2 such that ϕR1 = R2ϕ and ϕL1 = L2ϕ.

Analogously, a homomorphism ϕ : M1 → M2 of nonoriented maps M1 = (F1; λ1, ρ1, τ1) and M2 = (F2; λ2, ρ2, τ2) is a mapping ϕ : F1 → F2 such that ϕλ1 = λ2ϕ, ϕρ1 = ρ2ϕ and ϕτ1 = τ2ϕ.

3' 4'

1 2 1 2

4 3 4 3

2' 1'

Figure 3.6: Cube smoothly covering its halved quotient in the projective plane.

The properties of homomorphisms of both varieties of maps are similar except that homomorphisms of nonoriented maps ignore orientation. Every map homomorphism induces an epimorphism of the corresponding monodromy groups. Indeed, it is not diﬃcult to see that if ψ :(F1; λ1, ρ1, τ1) → (F2; λ2, ρ2, τ2)

26 R. Nedela: Group Actions on Surfaces

is a map homomorphism then the assignment λ1 7→ λ2, ρ1 7→ ρ2, τ1 7→ τ2 extends to an epimorphism ψ∗ called the induced epimorphism of the corresponding monodromy groups. Furthermore, transitive actions of the monodromy groups ensure that every map homomorphism is surjective and that it also induces an epimorphism of the underlying graphs. Topologically speaking, a map homomorphism is a graph preserving branched covering projection of the supporting surfaces with branch points possibly at vertices, face centers or free ends of semiedges. Therefore we can say that a map M˜ covers M if there is a homomorphism M˜ → M. A map homomorphism is smooth if it preserves the valency of vertices, the length of faces and does not send a link or a loop onto a semiedge. With map homomorphisms we use also isomorphisms and automorphisms. The automorphism group Aut (M) of an oriented map M = (D; R,L) consists of all permutations in the full symmetry group S(D) of D which commute with both R and L. Similarly, the automorphism group Aut (M) of an nonoriented map M = (F ; λ, ρ, τ) is formed by all permutations in the symmetry group Sym(F ) which commute with each of λ, ρ and τ. Hence, in both cases the automorphism group is nothing but the centralizer of the monodromy group in the full symmetry group of the supporting set of the map (cf. [91, Proposition 3.3(i)]). Since the action of the monodromy group Mon(M) is transitive, |Mon(M)| ≥ |D(M)| for every oriented map M, and |Mon(M)| ≥ |F (M)| for every nonoriented map M. If the equality is attained, then the monodromy group acts regularly on the supporting set, and therefore the map is called orientably-regular or regular, respectively. As it will become to be clear later the automorphism group of an orientably regular map M acts regularly on darts of M, and similarly Aut (M) of a regular map M acts regularly on flags of M. Our use of the term regular map thus agrees with that of Gardiner et al. [62] and Wilson [200], but is not yet standard. For instance, Jones and Thornton [92] uses the term “reflexible”, and White [192] calls such maps “reflexible symmetrical”. On the other hand, our orientably regular maps are called “regular” in Coxeter and Moser [51], “symmetrical” in [21] and [192], and “rotary” in Wilson [200]. For each homomorphism ϕ : M1 → M2 of oriented maps there is the cor- \ \ \ \ responding homomorphism ϕ : M1 → M2 defined by ϕ (x, i) = (ϕ(x), i). If M1 = M2 = M, that is, ϕ is an automorphism, then this definition and the assignment ϕ 7→ ϕ\ yield the isomorphic embedding of Aut (M) → Aut (M\). This allows us to treat Aut (M) as a subgroup of Aut\(M\) and, consequently, speak that every orientable regular map is orientably-regular (but not necessarily vice versa). It is easy to see that the index |Aut\(M\) : Aut (M)| is at most two. If it is two, then the map M is said to be reflexible, otherwise it is chiral. In the former case, there is an isomorphism ψ of the map M = (D; R,L) with −1 its mirror image (D; R ,L) called a reflection of M. Clearly, ψ\ is an automorphism that extends Aut (M) to Aut\(M\). Topologically speaking, oriented map automorphisms preserve the chosen orientation of the supporting surface whereas reflections reverse it. Transitivity of the action of the automorphism group of a regular (orientably regular) map forces all the vertices to have the same valency and all the faces to have same size (covalency). We say that a map M has a type (p, q) if the covalency of every face is p and the valency of every vertex of M is q for some

27 R. Nedela: Group Actions on Surfaces integers p, q. Generally, we can deﬁne the type of a map to be the couple (p, q) of integers, where p (q) is the least common multiple of covalencies (valencies) of the covalencies (valencies) of faces (vertices) of M.

Homomorphisms between regular maps Given homomorphism M → N between oriented maps more can be said if one of M, N , is a regular map. Let G ≤ Aut (M) = (D; R,L) be a subgroup of the automorphism group of an oriented map. For any x ∈ D denote [x] the orbit in the action of G containing x. Set R¯[x] = [Rx], L¯[x] = [Lx] for any x ∈ D and denote D¯ = {[x]; x ∈ D}. Then M¯ = (D¯; R,¯ L¯) is a well deﬁned oriented map and the natural assignment x 7→ [x] deﬁnes a map homomorphism M → M¯ . Homomorphisms arising in the above way are called regular, the group G ≤ Aut (M) is called the group of covering transformations. The following statement comes from [140].

Proposition 3.4 Let ϕ : M → N be a map homomorphism. If M is regular then ϕ is regular. In particular, any homomorphism between regular maps is regular, and moreover, in this case the group of covering transformations is a normal subgroup of Aut(M).

3.2 Generalization to hypermaps, Walsh map of a hypermap

A topological hypermap H is a cellular embedding of a connected trivalent graph X into a closed surface S such that the cells are 3-colored (say by black, grey and white colours) with adjacent cells having different colours. Numbering the colours 0, 1 and 2, and labeling the edges of X with the missing adjacent cell number, we can define 3 fixed points free involutory permutations ri, i = 0, 1, 2, on the set F of vertices of X; each ri switches the pairs of vertices connected by i-edges (edges labeled i). The elements of F are called flags and the group 1 G generated by r0, r1 and r2 will be called the monodromy group Mon(H) of the hypermap H. The cells of H colored 0, 1 and 2 are called the hypervertices, hyperedges and hyperfaces, respectively. Since the graph X is connected, the monodromy group acts transitively on F and orbits of hr0, r1i, hr1, r2i or hr0, r2i on F determine hyperfaces, hypervertices and hyperedges, respectively. The order of the element k = ord(r0r1), m = ord(r1r2) and n = ord(r2r0) is called the valency of a hyperface, hypervertex and hyperedge, respectively. The triple (k, m, n) is called type of the hypermap. 2 Maps correspond to hypermaps satisfying condition (r0r2) = 1, or in other words, maps are hypermaps of type (p, q, 2) or of type (p, p, 1). Thus we can view the category of Maps as a subcategory of the category of Hypermaps which is formed by 4-tuples (F ; r0, r1, r2), where ri (i = 0, 1, 2) are (fixed points free) involutory permutations generating the monodromy group Mon(H) acting transitively on F . Similarly, the category of Oriented Hypermaps arises by relaxing the condition L2 = 1 in the definition of an oriented map. More precisely, an oriented hypermap is a 3-tuple (D; R,L), where R and L are permutations

1This group has been called the monodromy group of H [96, 150], the connection group of H [200] and the Ω-group of H [26].

28 R. Nedela: Group Actions on Surfaces

(a) (b)

Figure 3.7: Regular embedding of the Fano plane in the torus gives rise to a regular hypermap of type (3, 3, 3). acting on D such that the oriented monodromy group is transitive on D. The notions deﬁned in the previous section extend from maps to hypermaps in an obvious way. For more information on hypermaps the reader is referred to [48]. The modiﬁed Euler formula for hypermaps reads as follows.

Theorem 3.5 (Euler formula for hypermaps) Let H = (F ; r0, r1, r2) be a hypermap on a closed surface S of genus g having v hypervertices, e hyperedges and f hyperfaces. Then (i) v + e + f − |F |/2 = 2 − 2g, if S is orientable, (ii) v + e + f − |F |/2 = 2 − g, if S is nonorientable.

Walsh representation An important and convenient way to visualize hypermaps was introduced by Walsh in [189]. Topologically, a map can be seen as a cellular embedding of a graph in a closed surface and a hypermap as a cellular embedding of a hy- pergraph in a closed surface. Since hypergraphs are in a sense bipartite graphs (with one monochromatic set of vertices representing the hypervertices and the other monochromatic set of vertices representing the hyperedges) a hypermap can be viewed as a bipartite map, as well. In fact, given any topological hypermap H we can construct a topological bipartite map W (H), called the Walsh bipartite map associated to H by taking ﬁrst the dual of the underlying 3- valent map and then deleting the vertices (together with the edges attached to them) lying inside the hyperfaces of H. The resulting map is bipartite with one monochromatic set of vertices lying on the faces colored black, representing the hypervertices of H, and the other monochromatic set lying on the faces colored grey, representing the hyperedges. This construction can be reversed: given any topological bipartite map B, where the vertices are bipartitioned in black and grey, we construct an associated topological hypermap W −1(B) = Tr(B∗) by truncating the dual map B∗; the faces of the resulting 3-valent map Tr(B∗) contains the vertices and the face- centres of the original map and are henceforth 3-colorable black, grey and white, with all these colours meeting at each vertex of Tr(M∗). If B = W (H) is the Walsh bipartite map of an oriented hypermap H = (D; R,L) then R and L are

29 R. Nedela: Group Actions on Surfaces

Figure 3.8: The Walsh map of the Fano plane embedding. the respective rotations on the two bipartition sets of the dart set of B, so the rotation of B is RL = LR.

3.3 Maps, hypermaps and groups Schreier representations In the previous section we have seen that maps and hypermaps can be represented by means of two or three permutations satisfying some conditions. The aim of this section is to show that one can study hypermaps as purely group theoretical objects. The idea emerges from the fact that every transitive permutation group is equivalent to a group acting on cosets by translation. Following [183, 184], we call these representations Schreier representations. Schreier representations of oriented maps appear implicitly in Jones and Singerman [91]. Vince [183] developed a theory of Schreier representations of (hyper)maps on closed surfaces described by three involutions. Here we introduce Schreier representations of oriented hypermaps. Let G be a finite group generated by two elements r and `. In other words, G is a finite quotient of some triangle group T +(k, m, n) = hr, `; `n = rm = (r`)k = 1i. k,m and n being positive integers. Further let S be a subgroup of G. Using the action of G on the set C = G/S of left cosets of S in G by the left translation, we construct a hypermap A(G/S; r, `) whose monodromy group is a homomorphic image of G and the local monodromy group is a homomorphic image of S. We take the cosets as darts of the hypermap and define the rotation R and the dart reversing involution L by setting

R(hS) = rhS, (3.1) L(hS) = `hS, (3.2) respectively, hS being an arbitrary element of C. For the resulting hypermap (C; R,L) = A(G/S; r, `) we easily check that the assignment r 7→ R, ` 7→ L extends to a homomorphism T +(k, m, n) → Mon(A(G/S; r, `)). Let H be a hypermap. A Schreier representation of H is an isomorphism H → A(G/S; r, `) for an appropriate group G = hr, ì and a subgroup S ≤ G, or simply the hypermap A(G/S; r, `) itself. Given any hypermap H = (D; R,L), it is not difficult to find a Schreier representation for H. Indeed, we first fix any dart a of H and set G = Mon(H) = hR,Li and S = Mon(H, a), to be

30 R. Nedela: Group Actions on Surfaces the stabilizer of a. Then, for an arbitrary dart x we take any element h ∈ H with h(a) = x and label x by the coset hS ∈ C, thereby obtaining a labeling α(x) = hS. Observe that α is well-defined since for any two elements h and h0 of Mon(M) with h(a) = x = h0(a) we have hS = h0S. In fact, α is a bijection of D(H) onto C. Clearly, α(Rx) = Rα(x) and α(Lx) = Lα(x) which means that α : H → A(G/S; R,L) is the required isomorphism. If we start from a given hypermap H, the Schreier representation we have just described is in some sense best possible because the monodromy group Mon(H) is not merely a homomorphic image of G but is actually isomorphic to it. In this case we say that the Schreier representation is effective. In general, a Schreier representation A(G/S; r, `) is effective if and only if G acts faithfully on C, i.e., the translation by every non-identity element of G is a non-identity permutation of C. Elementary theory of group actions or straightforwardT computations yield that the latter occurs precisely when the subgroup hSh−1, the core of S in h∈G G, is trivial (cf. Rotman [160]). For an arbitrary Schreier representation A(G/S; r, `) of a hypermap H we ∼ have Aut (H) = NG(S)/S, where NG(S) is the normalizer of S in G (see Proposition 3.7 in [50]). In particular, if H is orientably regular we can take G = Mon(H) and S = 1. Then Aut (A(G/1; r, `)) =∼ G =∼ Mon(A(G/1; r, `)), implying that Aut (H) =∼ Mon(H). Let us remark that the above isomorphism assigns the left translation by an element h ∈ G (representing a monodromy of H) to the right translation ξh (representing an automorphism of H). Summing up we get the following theorem. Theorem 3.6 Let H = (D; R,L) be an oriented hypermap. Then |Aut (H)| ≤ |D| ≤ |Mon(H)| and the following conditions are equivalent: (i) H is orientably regular, (ii) Mon(H) =∼ Aut (H), (iii) the action of Aut (H) on D is regular. In order to get a similar characterization of regular (nonoriented) hypermaps it suffices to replace, in the above statement, an oriented hypermap by a hypermap and darts by flags. Schreier representations provide a convenient tool to deal not only with automorphisms but also with homomorphisms between hypermaps. If G = hr, `; `n = rm = (r`)k = 1,... i is a finite quotient of the triangle group T +(k, m, n) and S ≤ S0 ≤ G are two subgroups then the natural projection π : G/S → G/S0, hS 7→ hS0 (h ∈ G), is a homomorphism A(G/S; r, `) → A(G/S0; r, `). In fact, every hypermap homomorphism ϕ : H1 → H2, where Hi = (Di; Ri,Li), is in the usual sense equivalent to an appropriate natural projection.

Generic hypermap One consequence of these considerations is that every oriented hypermap is a quotient of a (ﬁnite) oriented regular hypermap. In fact, for every oriented hypermap H = (D; R,L) there exists a regular hypermap H# and a

31 R. Nedela: Group Actions on Surfaces homomorphism π : H# → H with the following universal property: for every regular hypermap H˜ and a homomorphism ϕ : H˜ → H there is a homomorphism ϕ0 : H˜ → H# such that ϕ = πϕ0. In terms of Schreier representations, the homomorphism π is equivalent to the natural projection A(G/1; R,L) → A(G/S; R,L) =∼ H where G = Mon(H) and S = Mon(H, a), is the stabilizer of some dart a ∈ D(H). We shall call the hypermap H# the generic regular hypermap over H and π : H# → H the generic homomorphism. It is obvious that the induced homomorphism π∗ : Mon(H#) → Mon(H) is an isomorphism and that H# and H have the same type.

Construction of generic hypermap To construct the generic hypermap H# = (D#; R#,L#) for an oriented hypermap H = (D; R,L) it is suﬃcient to set D# = Mon(H), R#(x) = Rx, and L#(x) = Lx for any x ∈ D#. Observe that the automorphisms of H# are just the right translations of D# = Mon(H) by the elements of Mon(H), and so H# is indeed an orientably-regular hypermap. Similarly, if H = (F ; λ, ρ, τ) is an nonoriented hypermap then the generic regular hypermap H+ = (F +; λ+, ρ+, τ +) over H can be constructed by setting F + = Mon(H), λ+(x) = λx, ρ+(x) = ρx and τ +(x) = τx, for any x ∈ F +. Again, the hypermap automorphisms are given by the right translations of F + by the elements of Mon(H) = F +. It is obvious that if H is orientable, then so is H+. Moreover, the hypermap \ H+, as a topological hypermap, smoothly covers (H#) . In the following sections we shall see that maps or Walsh bipartite hypermap representations combined with the generic hypermap construction provide a convenient tool for construction regular hypermaps satisfying certain constrains (see for instance [90, 151]). Let us note that there is an interesting relationship between hypermaps and algebraic curves ( see Section 5). Via this relationship, actions of Galois groups of algebraic number ﬁelds on maps on surfaces are investigated (see [71, 96, 95, 165]). In this context the base maps are (following Grothendieck) called dessigns d’enfants.

Example 3.7 Figure 3.9 shows spherical maps M1,... M5 which generic maps are the ﬁve Platonic solids. The respective (oriented) monodromy groups are A4, S4, S4, A5 and A5. Dessign d’enfants M6 and M7 represent regular maps on surfaces with higher genera. The associated monodromy groups are the projective linear group PSL(2, 7) and Mathieu group M12, respectively. The generic map # M6 is known as the dual of the Klein’s triangulation of the surface of genus 3 and this map is the smallest Hurwitz map (see the following Section).

Maps and hypermaps from triangle groups The above theory of Schreier representations apply without any problem to inﬁnite hypermaps as well. It follows that oriented maps and hypermaps of given type (k, m, n) can be described as quotients of the universal oriented hypermap of type (k, m, n) which (oriented) monodromy group is T +(k, m, n). This is the even-word subgroup of the triangle group

2 2 2 k m n T (k, m, n) = hr0, r1, r2; r0 = r1 = r2 = (r1r2) = (r0r1) = (r2r0) = 1i,

32 R. Nedela: Group Actions on Surfaces

M1 M2 M3

M6 M7

Figure 3.9: Dessign d’enfants

which is the monodromy group of the universal hypermap A(T (k, m, n); r0, r1, r2) for the category of (nonoriented) hypermaps of type (k, m, n). It follows that

T +(k, m, n) = hR,L;(RL)k = Ln = Rm = 1i.

Note that the universal maps of type (k, m) with the monodromy group T (k, m, 2) are the well known tessellations of the sphere, plane or hyperbolic plane by 1 1 k-gons (m of them meeting at each vertex) provided the expression k + m 1 is greater, equal or less than 2 , respectively. It is well-known that a free 2- generator group can be represented as a matrix-group [159, pages 48-49]. Denote by ν = 2cos(π/k), η = 2cos(π/m) and ξ = 2cos(π/n). Set   1 ξ νξ + η r =  0 −1 −ν  (3.3) 0 ν ν2 − 1   −1 −ξ 0 ` =  ξ ξ2 − 1 0  . (3.4) η ηξ + ν 1 Then the assignment R 7→ r and L 7→ ` extends to a group monomorphism. Hence triangle groups are matrix groups, see [171, 172, 1] for more details. It follows that (hyper)maps can viewed as quotients of certain matrix groups, while regular (hyper)maps correspond to factor groups of the matrix groups representing the respective triangle groups. This approach to maps and hypermaps is used to study regular maps and hypermaps of large planar width, see Section 3.4.

33 R. Nedela: Group Actions on Surfaces

Hypermap subgroups We can go even one step further. Let us denote by

2 2 2 G = T (∞, ∞, ∞) = hr0, r1, r2; r0 = r1 = r2 = 1i, the free product of three two-element groups. Since the monodromy group of any hypermap H is a finite quotient of G we can identify every hypermap with the algebraic hypermap A(G/S; r0, r1, r2) for some S ≤ G of finite index. The subgroup S is called the hypermap subgroup. Consequently, one can study hypermaps via the subgroups of G of finite index. The facts listed in the following statement are well-known between map- and hypermap experts (see [48, 50]).

2 Theorem 3.8 Let H, H1 and H2 be hypermaps, and let G = hr0, r1, r2; r0 = 2 2 r1 = r2 = 1i. ∼ (a) H1 covers H2 if and only if there are S1 ≤ S2 ≤ G such that H1 = ∼ A(G/S1; r0, r1, r2) and H2 = A(G/S2; r0, r1, r2), ∼ (b) H1 = H2 if and only if the corresponding hypermap subgroups are conjugate in G, (c) H is orientable if and only if its hypermap subgroup is contained in the even-word subgroup G+ ≤ G,

(d) the hypermap subgroup of the nonoriented generic hypermap for a hypermap given by hypermap subgroup S ≤ G is the largest normal subgroup contained in S. In particular, regular hypermaps correspond to normal subgroups of G.

Using the algebraic representation via hypermap subgroups one can handle many problems. For instance, it is straightforward that given two hypermaps H1, H2 with the respective subgroups S1, S2, the intersection S1 ∩S2 deﬁnes the smallest common cover for both H1 and H2. Because of many advantages the investigation of maps and hypermaps via the corresponding hypermap subgroups posses, sometimes a hypermap itself is identiﬁed with its hypermap subgroup (see [15, 61]).

Product of hypermaps

The smallest common cover of given hypermaps H1 = (F1; r0, r1, r2) and H2 = (F2; q0, q1, q2) can be viewed as a product of two hypermaps. The question arises whether we can construct the product explicitly, or in other words, can we derive the monodromy group of the product in terms of the monodromy groups of factors? The most natural approach is to set F = F1 × F2 and pi(x, y) = (rix, qiy) for any (x, y) ∈ F and i = 0, 1, 2. Unfortunately, the hypermap H = (F ; p0, p1, p2) is, in general, not correctly deﬁned since the action of Mon(H1) × Mon(H2) may not be transitive on F . Necessary and suﬃcient condition ([33, 25]) follows.

Theorem 3.9 [33] Let H1, H2 be hypermaps with the hypermap subgroups S1 and S2. Then the monodromy group of the smallest common cover for H1, H2 is the direct product Mon(H1)×Mon(H2) if and only if T (∞, ∞, ∞) = hS1,S2i.

34 R. Nedela: Group Actions on Surfaces

The above theorem deals with a special instance of a general situation. Let H = U/H and K = U/K be algebraic hypermaps. Set H ∨ K = U/(H ∩ K) and H ∧ K = U/hH,Ki. The hypermaps H ∨ K, H ∧ K will be called join and intersection of H and K, respectively. If both hypermaps are regular then the hypermap subgroup is unique and join and intersection are well-defined binary operators in the category of regular hypermaps. Joins and intersections are studied in [29]. Join and intersection of regular hypermaps can be constructed using the following operation introduced by Wilson in [203]. Let A = hr0, . . . , rki and B = hs0, . . . , ski be two k-generated groups. Let us define their monodromy product A×m B to be the subgroup of the direct product generated by (ri, si), where i = 0, 1, . . . , k. Note that S. Wilson calls it the parallel product in [203]. Further, denote by π1 : A×m B → A, π2 : A×m B → B the natural projections erasing the second and first coordinate, respectively. The following theorem relates the above operations.

Theorem 3.10 [29] Let H = (A; r0, r1, r2) and K = (B; s0, s1, s2) be regular hypermaps. Then Mon(H ∨ K) = Mon(H) ×m Mon(K) and Mon(H ∧ K) = Mon(H) ×m Mon(K)/Ker π2Ker π1. Two regular hypermaps H, K will be called orthogonal if the respective hypermap subgroups generate T (∞, ∞, ∞) = hH,Ki. It follows that orthogonality of hypermaps is the necessary and suﬃcient condition under which the monodromy product is simple the direct product of the monodromy groups of factors. Joins and intersections of maps and hypermaps allow us to construct maps and hypermaps satisfying some special properties, for instance, self-dual or totally self-dual hypermaps, reﬂexible hypermaps. These operations play an important role in the study of the phenomenon of chirality which we are going to deal with in the next paragraph.

Chirality index Orientable hypermaps split into two families: family of reflexible and family of chiral hypermaps. Reflexible hypermap is mirror symmetric which means that the two associated oriented hypermaps (D; R,L) and (D; R−1,L−1) are isomorphic. Topologically speaking, a reflexible hypermap admits an orientation reversing self-homeomorphism of the supporting surface preserving the embedded graph and colours of faces. Note that S. Wilson [196] and some other authors use the term reflexible to denote (reflexible) regular hypermap. An orientable hypermap which is not reflexible is chiral (or mirror asymmetric). From the first point of view the “chirality” seems to be a binary invariant for the category of orientable hypermaps. Surprisingly, it turns out [32] that one can measure by a group, called the chirality group of a hypermap, of how much given orientable hypermap deviates from being mirror symmetric. For the simplicity let us restrict to orientably regular hypermaps now. Let us denote ∆ by H the largest (reflexible) regular hypermap covered by H and by H∆ the smallest (reflexible) regular hypermap covering H. It follows that H∆ = H∨Hr r r and H∆ = H ∧ H , where H denotes the mirror image of H. With the help of hypermap subgroup representation the following result is proved in [32].

Theorem 3.11 [32] Let H be an orientably regular hypermap. Then there exists a ﬁnite group G = X(H) of size κ such that H∆ is a smooth κ-fold cover of

35 R. Nedela: Group Actions on Surfaces

Figure 3.10: The smallest chiral orientably regular map is toroidal {4, 4}2,1

H, and H is a κ-fold cover of H∆. Moreover, both coverings are regular, with groups of covering transformations isomorphic to G.

The above deﬁned group X(H) is called the chirality group while its size κ is the chirality index. It follows that X(H) is trivial if and only if H is reﬂexible. Structure of chirality groups is studied in [32] in a more detail. It is proved there that every abelian group is the chirality group of an oriented regular hypermap. On the other hand, many non-abelian groups including symmetric groups and dihedral groups cannot serve as chirality groups.

Two-generator groups Finite groups generated by two involutions coincide with dihedral groups. Any finite 2-generator group can be interpreted as a monodromy group of a regular oriented hypermap, while groups generated by three involutions give rise to (nonoriented) regular hypermaps. A lot of finite groups belong to one or to the other above mentioned classes of groups. The following related problem can be found in the Kourovka notebook. Problem 3.1 [114, Problem 7.30] Characterize the finite simple groups generated by three involutions, two of which commute. Note that a regular map (F ; λ, ρ, τ) with a simple monodromy group is necessarily non-orientable. A similar problem was considered by Malle, Saxl and Weigel [134]. They proved that every non-abelian simple group can be generated by two elements, one of them being of order two. In other words, every non-abelian simple group is a monodromy group of some oriented regular map. Interesting problem of classification of all possible couples R,L (L2 = 1) of generators (up to the action of Aut (G)) for a given group G arises. Having in mind the above result, the solution of the problem for the class of (finite) simple groups means to prove a refinement of the classification of simple groups. Although, in general this problem seems to be intractable, at least for some groups it can be solved, see for instance Sah [161]. Moreover, using the Hall’s counting principle [75] one can calculate the number of nonisomorphic pairs of generators in terms of

36 R. Nedela: Group Actions on Surfaces group characters (see [59, 93]). Regular hypermaps which monodromy coincide with the monodromy groups of the ﬁve Platonic solids are classiﬁed in [27]

3.4 Regular maps of large planar width and residual ﬁniteness of triangle groups

While regularity (in either arithmetical or group-theoretical sense) is an obvious property of any generalization of the Platonic solids, it is somewhat less obvious whether or how their planarity should be generalized. The idea is to replace the global planarity of the Platonic solids by a certain local variant of this notion. This should guarantee that a suﬃciently “large” neighbourhood of each face is simply connected. Recent works in topological graph theory (cf. [145, 157, 158, 179]) suggest the following concept as a convenient measure of local planarity. A map M on a closed surface S other than the 2-sphere is said to have planar width at least k, w(M) ≥ k, if every non-contractible simple closed curve on S intersects the underlying graph of M in at least k points. Planar width (most often called “face-width” or “representativity”) has recently received a considerable attention as an important tool for the study of graph embeddings on surfaces [146, Chapter 5]. The following theorem presents the main result of [151]. Its proof consists in construction of a certain planar map Mw(p, q) of type (p, q) for which the generic map construction applies. Theorem 3.12 [151] For every pair of integers p ≥ 3 and q ≥ 3 such that 1/p + 1/q ≤ 1/2 and for every integer k ≥ 2 there exists an orientable regular map M of type (p, q) with planar width w(M) ≥ k. Moreover, we can require the map M to be reﬂexible. This theorem has several predecessors in the literature.

Grünbaum’s problem In 1976, Grünbaum [72] asked if for every pair of positive integers p and q with 1/p + 1/q < 1/2 (i.e., in the hyperbolic case) there are infinitely many finite regular maps of type (p, q). He also remarked, however, that it was not even known whether for such p and q there was at least one map of that type. The question was answered in the affirmative by Vince [184] (1983) within a more general framework of higher-dimensional analogues of regular maps. His proof, based on a theorem of Mal’cev saying that every finitely generated matrix group is residually finite (see, e.g., Kaplansky [103]), was non-elementary and non- constructive. Constructive proofs of Vince’s theorem were subsequently given by Gray and Wilson [63] and Wilson [199, 202] along with some refinements. Further constructions of regular maps of each type (p, q) have recently been given by Jendrol’ et al. [90] and Archdeacon et al. [6]. Perhaps the most elementary affirmative solution of the Grunbaum’s problem is obtained in [90] giving the dessing’s d’enfants for each hyperbolic type (p, q) and deriving the respective generic covering maps, see Figure 3.11, where p = qt + r, for some 0 ≤ r < q and t ≥ 1. Parallel to this development there is another line of research which is closely related to our main theorem. The bridge between the two streams is the observation that an orientable regular map of type (p, q) exists if and only if there is

37 R. Nedela: Group Actions on Surfaces

a)p−2 b)

q−r p−2

q−1 q−2 q−2 q−2 q−r−1

c) r

Figure 3.11: Dessigns d’enfants of hyperbolic types a ﬁnite group with presentation G = hx, y; xq = y2 = (xy)p = 1,... i forming the oriented monodromy group of the oriented regular map A(G; x, y). With this relationship in mind, the solution of the above Gr¨unbaum’s (p, q)-problem can be derived from an old result (1902) of Miller [144] (rediscovered by Fox [60] in 1952) which states:

Theorem 3.13 (Miller [144]) For any three integers p, q, and n, all greater than 1, there exist inﬁnitely many pairs of permutations α, β such that α has order p, β has order q, and αβ has order n, except that the three numbers are 2, 3, 3 or 2, 3, 4 or 2, 3, 5 (ordered arbitrarily) or two of the numbers are 2.

In the latter cases, the triples determine the groups uniquely: they are the tetra- hedral, the octahedral and the dodecahedral group, and the dihedral groups.

Hurwitz maps The special case where p = 7, q = 3 and n = 2 (or its dual, with the roles of p and q interchanged) was an object of an extensive research in the area of Fuchsian groups, hyperbolic geometry and Riemann surfaces originating from the famous theorem of Hurwitz (1893):

38 R. Nedela: Group Actions on Surfaces

Theorem 3.14 (Hurwitz’s Theorem) For any Riemann surface S of genus g ≥ 2, the number of its automorphisms (that is, conformal homeomorphisms) does not exceed 84(g − 1).

The Hurwitz bound is attained precisely when Aut (S) is a Hurwitz group, a finite group G generated by an element x of order 3 and an element y of order 2 whose product has order 7. As mentioned above, such a group gives rise to a regular map of type (7, 3) on S, a (trivalent) Hurwitz map, whose automorphism group is isomorphic to G. In this context, the fact that there are infinitely many Hurwitz groups was first established by MacBeath in 1969 [128]. More recent results on Hurwitz maps one can find in [39] where all Hurwitz groups giving rise to reflexible regular maps of type (7, 3) and of genus < 11, 905 are classified.

Figure 3.12: Barycentric subdivision of the universal Hurwitz map

An interesting problem considered by Conder and others reads as follows: Problem. Which simple groups are Hurwitz groups? It transpires that except for a finite family of alternating groups the answer was done in a serie of articles, see [38, 40, 44] for instance. Let us note that MacBeath’s theorem immediately follows from Vince’s theorem which in turn is a consequence of Theorem 3.12. Indeed, it is sufficient to take an infinite sequence of regular maps of any type (p, q) (in particular, (7, 3)) with increasing planar width. Thus Theorem 3.12 has the following two corollaries:

Theorem 3.15 (Vince [184]) For any pair of integers with p ≥ 2, q ≥ 2 and 1/p+1/q ≤ 1/2 there exist inﬁnitely many orientable regular maps of type (p, q).

39 R. Nedela: Group Actions on Surfaces

Theorem 3.16 (MacBeath [128]) There exist inﬁnitely many Hurwitz groups.

Residual finiteness of triangle groups Another notable consequence of Theorem 3.12 is of group-theoretic nature. Let T +(p, q, 2) be the oriented triangle group with presentation hx, y; xq = y2 = (xy)p = 1i. Then for any integer w ≥ 1 there exist infinitely many finite quotients H of T +(p, q, 2) such that in any presentation of H in terms of x and y all reduced identities that are not identities of T +(p, q, 2) have length greater than w. The latter sentence is nothing but a reformulation of the well-known fact that triangle groups T +(p, q, 2) are residually finite [103]. A group G is called residually finite if for each element g ∈ G there exists a finite quotient H of G such that the epimorphism G → H does not send g onto the identity. Furthermore, as it was already mentioned the group T +(p, q, 2) is isomorphic to the even-word subgroup of the full triangle group T (p, q, 2) = hx, y, z; x2 = y2 = z2 = (xy)p = (yz)2 = (xz)qi. Hence, by using Theorem 3.12 (as well as its nonoriented version) we deduce the following corollary (See also [172, 173].)

Theorem 3.17 For each pair of integers p and q such that 1/p+1/q ≤ 1/2 both the oriented and the full triangle group T +(p, q, 2) and T (p, q, 2) are residually ﬁnite.

As was noted, Vince [184] proved Grünbaum’s conjecture by employing the residual finiteness of triangle groups. In contrast, we have just shown that Theorem 3.12 implies both Vince’s theorem and the residual finiteness of triangle groups. In fact, the residual finiteness of triangle groups is equivalent to Theorem 3.12. It is well known (see [79, 127, 162]) that the fundamental groups of closed surfaces are residually finite. Alternatively, this can be proved by using Corol- lary 12 and observing that every fundamental group of a closed surface embeds into some triangle group T (p, q, 2). A natural way to make see that the fundamental group of a given surface embeds into a triangle group T (p, q, 2) is to try to construct a map of type (p, q) supported by S. For instance, if the surface is orientable of genus g one can take p = 4g = q. In [90, 151, 172, 173] one can find further applications of Theorem 3.12. The constructions of regular maps of large planar width based on construction of generic maps over some planar maps produce maps with enormously large size. Generalizing the works of Siráˇn[172,ˇ 173], Abas [1] proved the following statement giving much better bounds. It represents the strongest result related to residual finiteness of triangle groups.

Theorem 3.18 Let 1/k + 1/m + 1/n < 1. Then for every w ≥ 1 there exists a regular oriented hypermap (D; R,L) with planar width at least w such that the 2 2 2 |D| ≤ Cw, where C = C(k, m, n) < 29kmn((k +m +n )−(k+m+n)+3).

3.5 Maps, hypermaps and Riemann surfaces

The aim of this section is to explain brieﬂy the relationship between hypermaps and Riemann surfaces. Following the approach of Jones [96] we ﬁrstly show that any hypermap can be endowed with the structure of Riemann surface. A

40 R. Nedela: Group Actions on Surfaces natural question arises: What kind of Riemann surfaces are associated with hypermaps? Surprisingly beautiful answer [96] is a consequence of the theorem of Bely˘ı[11]. In what follows we have extracted some ideas from [96] where one can find more detailed information as well as an exhaustive list of relevant references. A Riemann surface is a surface with locally-defined complex coordinates, such that the changes of coordinates between intersecting neighbourhoods are conformal. The the upper half-plane U = { z ∈ C | Im z > 0 }, and the Riemann sphere (or complex projective line) Σ = P 1(C) = C ∪ {∞} are examples of simply-connected Riemann surfaces. The upper half-plane is a model of hyperbolic geometry, the geodesics being the euclidean lines and semi-circles which meet the real line R at right-angles. The modular group Γ = PSL2(Z), consisting of the Möbiustransformations az + b T : z 7→ (a, b, c, d ∈ Z, ad − bc = 1), cz + d acts on U as a group of orientation-preserving hyperbolic isometries. It is well- known (see for instance [96]) that modular group is a free group of rank 2. Hence it can serve as a universal covering group for the set of monodromy groups of oriented hypermaps which are two-generator groups. In order to get a representation of a hypermap by means of a Riemann surface one has to embed the underlying graph of the universal Walsh bipartite map into U in the “right way”. This can be done provided we extend U as follows. Observe that Γ acts (transitively) on the rational projective line P 1(Q) = Q ∪ {∞}, and hence it acts on the extended hyperbolic plane

U = U ∪ Q ∪ {∞} .

Let us denote by a [0] = { ∈ Q ∪ {∞} | a is even and b is odd } , b a [1] = { ∈ Q ∪ {∞} | a and b are both odd } . b The subgroup stabilizing both [0] and [1] is

Γ(2) = { T ∈ Γ | b ≡ c ≡ 0 mod (2) } .

The universal (Walsh) bipartite map Bˆ on U has [0] and [1] as its sets of black and white vertices, and its edges are the hyperbolic geodesics between vertices a/b and c/d where ad−bc = ±1; this implies that a and c have opposite parity, so the map is indeed bipartite. The automorphism group of Bˆ (preserving orientation and colours) is Γ(2). This is a free group of rank 2, freely generated by z z − 2 Rˆ : z 7→ and Lˆ : z 7→ . −2z + 1 2z − 3 It follows that if B is any bipartite map, representing an oriented hypermap H with monodromy group G = hR,Li, then there is an epimorphism

Γ(2) → G, Rˆ 7→ R, Lˆ 7→ L,

41 R. Nedela: Group Actions on Surfaces giving a transitive action of Γ(2) on the set E of edges of B. The stabilizer of an edge in this action is a subgroup B of index N = |E| in Γ(2), called the map subgroup corresponding to B (diﬀerent choices of an edge lead to conjugate subgroups). Since B ≤ Γ(2) = Aut(Bˆ), one can form the quotient map Bˆ/B isomorphic to B. In particular, the oriented hypermap H is regular if and only if B is normal in Γ(2), in which case Aut (H) =∼ Γ(2)/B =∼ G. Note that the above map subgroup B can be viewed as an oriented version of the hypermap subgroup introduced above.

−1/3 −1/5 0 1/5 1/3 2/5 3/5 2/3 4/5 1 6/5 4/3

Figure 3.13: Universal hypermap on the extended upper-half plane.

Using the above factorization process we end with an isomorphic copy of our original Walsh bipartite map B, endowed with some extra structure. The underlying surface is now a compact Riemann surface X = U/B, in which the underlying graph is very rigidly embedded: the edges are all geodesics, the angles between successive edges around a vertex are all equal, and the automorphisms of B are all conformal automorphisms of U/B (induced by the action of NΓ(2)(B) on Bˆ). Clearly, we can represent the trivial bipartite map on the Riemann surface Σ with one black vertex at 0, one white vertex at 1, and the unique edge joining them as the interval [0,1]. Let B∞ denotes this representation. Since any bipartite map covers B∞, alternatively, one can derive the structure of Riemann surface associated with B by considering the branched regular covering ∼ ∼ B = Bˆ/B → B1 = Bˆ/Γ(2) given by the inclusion B ≤ Γ(2).

Riemann surfaces and algebraic curves If A(x, y) ∈ C[x, y] is a polynomial in x and y with complex coeﬃcients, then the equation A(x, y) = 0 deﬁnes the complex variable y as an N-valued function of the complex variable x, where N is the degree of A in y. Consequently, the Riemann surface XA of this equation can be constructed by taking N copies of the Riemann sphere Σ (one for each branch of the function), cutting them between the branch-points, and then rejoining the sheets across these cuts. A Riemann surface is called algebraic if it is isomorphic to XA for such a polynomial A. The following major result was known to Riemann:

Theorem 3.19 (Riemann) A Riemann surface is compact if and only if it is algebraic.

If K is a subfield of C, then we say that a compact Riemann surface X is ∼ defined over K if X = XA for some polynomial A(x, y) ∈ K[x, y]. Let Q denotes the field of algebraic numbers. A Bely˘ıfunction is a meromorphic function X → Σ with no critical values outside {0, 1, ∞}. The following powerful result is due to Bely˘ı[11]:

42 R. Nedela: Group Actions on Surfaces

Theorem 3.20 (Bely˘ı) A compact Riemann surface X is deﬁned over Q if and only if there is a Bely˘ıfunction β : X → Σ.

Bely˘ıtheorem implies the following theorem (see Jones [96]) giving a correspondence between hypermaps, Riemann surfaces and algebraic curves. It shows that the Riemann surfaces associated with hypermaps are precisely those deﬁned over the ﬁeld of algebraic numbers.

Theorem 3.21 [96] If X is a compact Riemann surface then the following are equivalent:

X is deﬁned over Q;

X =∼ U/M for some subgroup M of ﬁnite index in the modular group Γ; X =∼ U/B for some subgroup B of ﬁnite index in Γ(2);

X =∼ U/H for some subgroup H of ﬁnite index in a hyperbolic triangle group T (k, m, n) where the integers k, m, n satisfy 1/k + 1/m + 1/n < 1.

3.6 Enumeration of maps of given genus Map enumeration problem Enumeration of maps on surfaces has attracted a lot of attention last decades. In enumeration problems we are mostly interested in enumeration of maps without semi-edges. When needed to stress it we will call such maps ordinary maps. As shown in monograph [125] the enumeration problem was investigated for various classes of maps. Generally, problems of the following sort are considered:

Problem 3.2 How many isomorphism classes of maps of given property P and given number of edges (vertices, faces) there are? Beginnings of the enumerative theory of maps are closely related with the enumeration of plane trees considered in 60-th by Tutte [181], Harary, Prins and Tutte [76], see [77, 124] as well. Later a lot of other distinguished classes of maps including triangulations, outerplanar, cubic, Eulerian, nonseparable, simple, looples, two-face maps e.t.c. were considered. Research in these areas till year 1998 is well represented in [125]. Although there are more than 100 published papers on map enumeration, see for instance [14, 23, 49, 112, 120, 188, 190, 207], most of them deal with the enumeration of rooted maps of given property. In particular, there is a lack of results on enumeration of unrooted maps of genus ≥ 1. The presented results can be viewed as an attempt to ﬁll in this gap. The problem considered in this section reads as follows.

Problem 3.3 What is the number of isomorphism classes of oriented unrooted maps of given genus g and given number of edges e?

An oriented map is called rooted of one of the darts (arcs) is distinguished as a root. Isomorphisms between oriented rooted maps take root onto root. A rooted variant of Problem 3.3 follows.

43 R. Nedela: Group Actions on Surfaces

Problem 3.4 What is the number of isomorphism classes of oriented rooted maps of given genus g and given number of edges e? Problem 3.4 was first considered in 1963 by Tutte [180] for g = 0, i.e. for the planar case. The corresponding planar case of the unrooted version (Problem 3.3 for g = 0) was settled by Liskovets [119, 121] and Wormald [206] much later. The enumeration problem for maps of given genus g > 0 and given number of edges was considered by Walsh and Lehman in [187, 188]. They have derived an algorithm based on a recursion formula. The algorithm is applied to enumerate maps with small number of edges. An explicit formula for number of rooted maps for g = 1 is obtained by D. Arquès[8]. In 1988 Bender, Canfield and Robinson [12] derived an explicit enumeration formula for the number of rooted maps on torus and projective plane. Three years later [13] Bender and Canfield determined the function Ng(e) of rooted maps of genus g with e edges for any genus g up to some constants. For g = 2 and g = 3 the generating functions are derived. Some refinement of these results can be found in [9]. In what follows we shall deal with the problem of enumeration of oriented unrooted maps with given genus and given number of edges. Inspired by a fruitful concept of an orbifold used in low dimensional topology and in theory of Riemann surfaces we introduce a concept of a map on an orbifold. As it will become clear to be later, cyclic orbifolds, that is the quotients of the type Sγ /Z`, where Sγ is an orientable surface of genus γ surface and Z` is a cyclic group of automorphisms of Sγ , will play a crucial role in the enumeration problem. The enumeration formula allows us to decompose the problem into two subproblems (see Theorem 3.23). First one requires an enumeration of certain epimorphisms defined on Fuchsian groups (or on F -groups) onto a cyclic group. The other requires to enumerate rooted maps on cyclic orbifolds associated with the considered surface. Unfortunately, quotients of (ordinary) maps may have semiedges. We show how to reduce the problem of enumeration of rooted maps with semiedges sitting on orbifolds to the problem of enumeration of ordinary rooted maps on surfaces.

Maps on oriented orbifolds Given regular covering ψ : M → N , let x ∈ V (N ) ∪ F (N ) ∪ E(N ) be a vertex, face or edge of N . The ratio of degrees b(x) = deg(˜x)/deg(x), where x˜ ∈ ψ−1(x) is a lift of x along ψ, will be called a branch index of x. The branch index is a well-defined positive integer not depending on the choice of the liftx ˜. In some considerations, it is important to save information about branch indexes coming from some regular covering defined over a map N. This can be done by introducing a signature σ on M.A signature is a function σ : x ∈ V (N ) ∪ F (N ) ∪ E(N ) → Z+ assigning a positive integer to each vertex, edge and face, with the only restriction: if x is an edge of degree 2 then σ(x) = 1, if it is of degree 1 then σ(x) ∈ {1, 2}. We say that a signature σ on N is induced by a covering ψ : M → N if it assigns to vertices, faces and edges of N their branch indexes with respect to ψ. Given a couple (M, σ), where M is a finite oriented map and σ is a signature we define an orbifold type of (M, σ) to be an (r + 1)-tuple of the form [g; m1, m2, . . . , mr], where g is the genus of the underlying surface, 1 < m1 ≤

44 R. Nedela: Group Actions on Surfaces

m2 ≤ · · · ≤ mr are integers, and mi appears in the sequence si > 0 times if and only if σ takes the value mi exactly si times. The orbifold fundamental group π1(M, σ) of (M, σ) is a F-group

π1(M, σ) = F [g; m1, m2, . . . , mr] =

Yg Yr m1 mr ha1, b1, a2, b2, . . . , ag, bg, e1, . . . , er| [ai, bi] ej = 1, e1 = . . . er = 1i. i=1 j=1 In particular, the oriented triangle group T +(k, m, n) is the F -group F [0; k, m, n]. Let ψ : M → N be a regular covering and σ be a signature defined on N . We say that ψ is σ-compatible if for each element x ∈ V (N ) ∪ E(N ) ∪ F (N ) the branch index of b(x) of x is a divisor of σ(x). Signature σ defined on N lifts along a σ-compatible regular covering ψ : M → N to a derived signature σψ −1 on M defined by the following rule: σψ(˜x) = σ(x)/b(x) for eachx ˜ ∈ ψ (x) and each x ∈ V (N ) ∪ E(N ) ∪ F (N ). Let us remark that if σ(x) = 1 for each x ∈ V (N ) ∪ E(N ) ∪ F (N ) then σ-compatible covers over M are just smooth regular covers over M. Such a signature will be called trivial. The following well-known statement relates the signature induced by a regular covering between two oriented maps with the genera of the respective surfaces.

Theorem 3.22 (Riemann-Hurwitz equation) Let M → M/G be a regular covering between maps with a covering transformation group G, let M be ﬁnite. Let the respective orbifold type of N = M/G be [g; m1, m2, . . . , mr]. Then the Euler characteristic of the underlying surface of M is given by the

¡ Xr 1 ¢ χ = |G| 2 − 2g − (1 − ) . m i=1 i A topological counterpart of a (combinatorial) map M with a signature σ can be established as follows. By an orbifold O we mean a surface S with a distinguished discrete set of points B assigned by integers m1, m2, . . . , mi,... such that mi ≥ 2, for i = 1, 2,... . Elements of B will be called branch points. If S is a closed compact connected orientable surface of genus g then B is ﬁnite of cardinality |B| = r and O is determined by its type [g; m1, m2, . . . , mr]. Hence we write O = O[g; m1, m2, . . . , mr]. The fundamental group π1(O) of O is an F -group deﬁned above. A (topological) map on an orbifold O is a map on the underlying surface Sg of genus g satisfying the following properties: (P1) if x ∈ B then x is either an internal point of a face, or a vertex, or an end-point of a semiedge which is not a vertex, (P2) each face contains at most one branch point, (P3) the branch index of x lying at the free end of a semiedge is two. A mapping ψ : O˜ → O is a covering if it is a branched covering between underlying surfaces mapping the set of branch points B˜ of O˜ onto the set B of −1 branch points of O and eachx ˜i ∈ ψ (xi) is mapped uniformly with the same branch index d dividing the prescribed index ri of xi ∈ B.

45 R. Nedela: Group Actions on Surfaces

Enumeration formula

Let Sγ be an orientable surface of genus γ. Denote by Orb(Sγ /Z`) the set of cyclic orbifolds given by all actions of the cyclic group Z` of order ` on Sγ . Given orbifold O denote by Epi0(π1(O),Z`) the number of epimorphisms from the orbifold fundamental group π1(O) onto the cyclic group Z` with a torsion- free kernel. For a cyclic orbifold O ∈ Orb(Sγ /Z`) denote by µO(m) the number of rooted maps with m darts on O. The following formula is proved in [142]. Theorem 3.23 [142] With the above notation the number of ordinary oriented maps of genus γ having e edges is 1 X X ν (e) = Epi (π (O),Z ) · µ (m). γ 2e 0 1 ` O `|2e O∈Orb(Sγ /Z`) 2e = `m

The following two propositions determine the functions Epi0(π1(O),Z`), µO(m).

Proposition 3.24 [142] Let Γ = F [g; m1, . . . , mr] be an F −group of signature (g; m1, . . . , mr). Denote by m = lcm (m1, . . . , mr) the least common multiple of m1, . . . , mr and let m|`. Then the number of order-preserving epimorphisms of the group Γ onto a cyclic group Z` is given by the formula

2g Epi0(Γ, Z`) = m φ2g(`/m)E(m1, m2, . . . , mr), where 1 Xm E(m , m , . . . , m ) = Φ(k, m ) · Φ(k, m ) ... Φ(k, m ), 1 2 r m 1 2 r k=1

φ2g(`) is the Jordan multiplicative function of order 2g, and Φ(k, mj) is the von Sterneck function. In particular, if Γ = F [g; ∅] = F [g; 1] is a surface group of genus g we have

Epi0(Γ, Z`) = φ2g(`).

Recall that Ng(e) is the number of ordinary rooted maps of genus g having e edges. A signature of an orbifold O ∈ Orb(Sγ /Z`) can be written in an exponential form as follows [g; 2q2 , . . . , `q` ]. The symbol iqi means: the number of points with branch index i is qi.

q2 q Proposition 3.25 [142] Let O = O[g; 2 , . . . , ` ` ] be an orbifold, qi ≥ 0 for i = 2, . . . , `. Then the number of rooted maps νO(m) with m darts on the orbifold O is

q µ ¶µ ¶ X2 m m−s +2−2g ν (m) = 2 N ((m−s)/2), O s q −s, q , . . . , q g s=0 2 3 ` with a convention that Ng(n) = 0 if n is not an integer. The above results have a group theoretical interpretation. The enumeration of ordinary rooted maps of given genus and number of edges is equivalent to the enumeration of torsion-free subgroups of given genus and index of the free prod- ∼ + uct Z ∗ Z2 = T (∞, ∞, 2), while the enumeration of ordinary unrooted maps

46 R. Nedela: Group Actions on Surfaces is the same as the enumeration of conjugacy classes of torsion-free subgroups of Z ∗ Z2 of given genus and index. The following list containing the numbers of rooted and oriented unrooted maps of genus 1 up to 30 edges follows. The numbers of toroidal unrooted maps were obtained using Theorem 3.23 with the help of software MATHEMATICA 4.1.

No. edges, No. rooted maps on torus, No. unrooted maps on torus:

02, 1, 1 03, 20, 6 04, 307, 46 05, 4280, 452 06, 56914, 4852 07, 736568, 52972 08, 9370183, 587047 09, 117822512, 6550808 10, 1469283166, 73483256 11, 18210135416, 827801468 12, 224636864830, 9360123740 13, 2760899996816, 106189359544 14, 33833099832484, 1208328304864 15, 413610917006000, 13787042250528 16, 5046403030066927, 157700137398689 17, 61468359153954656, 1807893066408464 18, 747672504476150374, 20768681225892328 19, 9083423595292949240, 239037464947999900 20, 110239596847544663002, 2755989928117365244 21, 1336700736225591436496, 31826208029615881656 22, 16195256987701502444284, 368074022535205870382 23, 196082659434035163992720, 4262666509741017440552 24, 2372588693872584957422422, 49428931123444048643388 25, 28692390789135657427179680, 573847815786545413529104 26, 346814241363774726576771244, 6669504641624799675973078 27, 4190197092308320889669166128, 77596242450201993985513136 28, 50605520500653135912761192668 903670008940406050891508432 29, 610946861846663952302648987552 10533566583563768540393559344 30, 7373356726039234245335035186504 122889278767322703855171530872 Let us remark that the initial values conﬁrm the available data for e ≤ 6 obtained by Walsh [188] (the sequence M4253 in [170]). Similar computations were done for surfaces up to genus three [142]. For larger genera both the numbers of rooted maps and unrooted maps are, except some small values of e, not determined.

47 R. Nedela: Group Actions on Surfaces

3.7 Regular hypermaps on a fixed surface Orientably regular hypermaps of genus at most two By the Hurwitz bound (see Theorem 3.14) the size of the automorphism group, and consequently, the size of the oriented monodromy group G = hR,Li of an orientably regular hypermap H = (D; R,L) is bounded by 84(g − 1), where g > 1 is the genus. Hence, a surface of genus at least two admits only finitely many orientably regular hypermaps. One of the central problems in the theory of maps and hypermaps is the problem of classification of all orientably regular hypermaps on a fixed underlying surface. Problem 3.5 Classify (orientably) regular hypermaps on a given surface. Recall that orientable regular hypermaps of genus g form a subset of the orientably regular hypermaps of genus g. Also the classification of regular hypermaps on a non-orientable surface of genusg ˜ can be derived from the classification of orientably regular hypermaps of genus g =g ˜ − 1 by using the construction of the antipodal cover. Hence the classification problem for regular hypermaps on non-orientable surface of genusg ˜ can be solved by checking the list of all orientably regular hypermaps of genus g =g ˜ − 1 (see [150] for more detailed explanation). Using the Euler formula (see Theorems 3.1, 3.5) it is not difficult to see that spherical orientably regular hypermaps consist of the five Platonic solids, and of two infinite families of types (1, n, n), (2, 2, n) and their duals. Orientably regular maps on torus were classified by Coxeter and Moser [51], and the generalization to hypermaps was done by Corn and Singerman [50]. Up to duality, there are three infinite families of toroidal orientably regular hypermaps which types are (2, 4, 4), (2, 3, 6) and (3, 3, 3), see Fig. 3.14. In [26, 28] the classification problem for double torus is settled. As concerns surfaces of higher genera only partial results for hypermaps are known. For instance, in [30] mirror asymmetric orientably regular hypermaps up to genus 4 are classified.

Regular maps with prescribed genus In this subsection we shall discuss the following problem. Problem 3.6 Given surface S classify (oriented) regular maps on S. Sherk in [163] classified oriented regular maps of genus 3. Grek in [64, 65, 66, 67] derived characterization of nonoriented and oriented regular maps on surfaces (orientable and nonorientable) with Euler characteristic ≥ −4. Regular maps and oriented regular maps on surfaces of genus less than 8 and non-orientable surfaces of genus less than 9 have been completely classified by Garbe (see [61] and [15]). Recently, Conder and Dobcsányi [43] with the help of a computer programme gave a list of all orientably regular maps up to genus 15.

Existence problem Following MacLachlan [135] (see also [2]) let us denote by µ(g) the order of largest group of conformal self-mappings (automorphisms) of a compact Rie- mann surface of genus g. Hurwitz bound gives µ(g) ≤ 84(g − 1). The following result was proved independently by Accola [2] and MacLachlan [135].

48 R. Nedela: Group Actions on Surfaces

(-c+2b;b+2c)

(-c+b;b+c)

(-c;b) (2b;2c)

b (b;c) l2

c c l1

c O b XX

Figure 3.14: Toroidal regular map {4, 4}b,c as a (regular) quotient of the universal map of type {4, 4}

Theorem 3.26 [135, 2] With the above notation we have 8(g + 1) ≤ µ(g) ≤ 84(g − 1) provided g ≥ 2, There are inﬁnitely many integers g ≥ 2 for which the equality µ(g) = 8(g + 1) holds.

Having in mind the correspondence between hypermaps and Riemann surfaces we get that there are inﬁnitely many surfaces S such that the number of darts of any orientably regular hypermap on S is at most 8(g + 1). On the other hand, one needs at least 4g darts to cut the surface into cells. Thus the size of the monodromy group of an orientably regular map on such a surface ranges between 4g and 8(g + 1). While in the orientable case, for any g ≥ 1, the regular embedding of the bouquet of 2g cycles gives a regular map of genus g, search for regular maps on nonorientable surfaces shows that there are surfaces which do not support any regular map. More precisely, Breda and Wilson prove in [205] the following result.

Theorem 3.27 [205] Let g be an integer 1 ≤ g ≤ 52. Then there is a regular non-orientable hypermap of genus g if and only if g is diﬀerent from 2, 3, 18, 24, 27, 39 or 48.

On the other hand, Conder and Everitt [42] cover by constructions about 75 percent of non-orientable genera. The following problem can be found in [42]. Problem. Are there inﬁnitely many non-orientable surfaces supporting no regular (hyper)maps.

49 R. Nedela: Group Actions on Surfaces

The result proved in [34] represents a breakthrough in the nonorientable regular map classification problem. A complete classification of all nonorientable regular maps with negative odd prime Euler characteristic – or, equivalently, regular maps on nonorientable surfaces of genus p + 2 where p is an odd prime, is derived. As a by-product an affirmative answer to the above existence problem is obtained. To be able to state our main result in a condensed form, for p ≡ −1 (mod 4) we denote by ν(p) the number of pairs of positive integers (j, l) such that j > l ≥ 3, both j and l are odd, and (j − 1)(l − 1) = p + 1.

Theorem 3.28 [34] Let p be an odd prime, p 6= 7, 13, and let Np+2 be a nonorientable surface of Euler characteristic −p (and hence of genus p + 2).

(1) If p ≡ 1 (mod 12) then there is no regular map on Np+2. (2) If p ≡ 5 (mod 12) then, up to isomorphism and duality, there is exactly one regular map on Np+2. (3) If p ≡ −5 (mod 12) then, up to isomorphism and duality, there are ν(p) regular maps on Np+2.

(4) If p ≡ −1 (mod 12) then, up to isomorphism and duality, Np+2 supports exactly ν(p) + 1 regular maps.

Corollary 3.29 There are no regular maps of Euler characteristic −p, where p is a prime ≡ 1 mod 12, p > 13. In particular, there are inﬁnitely many surfaces supporting no regular maps.

Modifications of the classification and existence problem can be considered. For instance, in [30] the classification of chiral hypermaps with genus at most four is carried. The main result of [30] implies that there is no chiral (orientably regular) hypermap on surface of genus 2, while each of the surfaces of genus 3 and 4 supports (up to duality) exactly one chiral hypermap. Examining the list of orientably regular maps up to genus 15 [43] one can see that surfaces of genera 2,3,4,5,6,9 and 13 support no chiral maps.

Improving Hurwitz bound The above mentioned results of Accola, McLachlan, Conder and others, as well as Theorem 3.28 suggest that the Hurwitz bound 84(g − 1), (168(g − 1)) on the size of discrete group of automorphisms of a Riemann surface of genus g can improved when considering particular sets of surfaces. Recently published paper [45] contains a discussion on possible improvements of the Hurwitz bound. They deﬁne a group of automorphisms of a non-orientable compact surface large if its size is > 4(−χ). The Riemann-Hurwitz formula then severely restricts possible signatures of large groups. Non-trivial families of large groups correspond to regular maps and regular hypermaps of types (k, m, 3), k ≥ m ≥ 3; (k, m, 4), −χ > k ≥ m ≥ 4 and (k, m, 5), 9 ≥ k ≥ m ≥ 5. If −χ is prime the following theorem classiﬁes large groups that correspond to regular hypermaps.

Theorem 3.30 [102] Let G be an automorphism group of a regular hypermap of type (k, m, n), k ≥ m ≥ n ≥ 3 acting on a surface of characteristic χ = −p, where p is an odd prime. If k ≥ m ≥ n = 3, 4 or 5, then either

50 R. Nedela: Group Actions on Surfaces

∼ (i) G = L2(7) with p = 7 and [k, m, n] = [4, 3, 3] with a reduced presentation hr, s| r4 = s3 = (rs)3 = (rs−1)4i, or ∼ ∼ (ii) G = P GL2(5) = S5 with p = 23 and [k, m, n] = [6, 5, 4] with a reduced presentation hr, s| r6 = s5 = (rs)4 = 1, (rs−1)2 = rsr−2s2r−1s−1 = 1i. In each case, there is a unique action of G, up to equivalence.

A classiﬁcation result on large groups acting on orientable surfaces of genus g = p + 1, p a prime, was proved by Belolipetsky and Jones in [10]. There the set of large groups is deﬁned more restrictively requiring |G| > 6p.

3.8 Operations on maps and hypermaps, external symmetries of hypermaps

Generally, an operation Φ is a function associating a given (hyper)map M another (hyper)map Φ(M). Typically, we require that Φ preserve some important properties of maps such as the underlying surface, or the underlying graph, or the monodromy group, or the hypermap subgroup etc. With each operation Φ and a ﬁxed map M a family of external symmetries, deﬁned by the relation M =∼ Φ(M), is associated. Depending on the class of operations we consider, the external symmetries, called exomorphisms in [150], form a group Exo(M) containing the automorphism group Aut (M) as a (normal) subgroup. The factor group Exo(M)/Aut (M) then can be interpreted as a group of outer symmetries leaving M invariant.

Functors Important class of operations is formed by functors between categories of hypermaps, i.e., it is required that morphisms between hypermaps are preserved. Perhaps the most familiar functors in the category of nonoriented maps is the duality operation defined by (F ; λ, ρ, τ) 7→ (F ; τ, ρ, λ), and the Petrie operation defined by (F ; λ, ρ, τ) 7→ (F ; λ0, ρ, λ), where λ0(x) = τ(x) or λ0(x) = λτ(x) depending on whether a flag x is, or is not, associated with a semiedge, respectively. These two functors have a nice geometric description also. The above two functors generate a group of functors isomorphic to S3 (see Wilson [197] and Lins [118]). Jones and Thornton showed that the above group of six operations is induced by the action of the outer automorphism group on conjugacy classes of subgroups of the group T (∞, ∞, 2) = hλ, ρ, τ; ρ2 = τ 2 = λ2 = (τλ)2 = 1i, which is the automorphism group of the universal map for the category of nonoriented maps.

Theorem 3.31 [92] The set of operations acting on the family of (nonoriented) maps induced by the action of outer automorphism group of T (∞, ∞, 2) consists of six operations forming a group G generated by the duality and Petrie operations, and G is isomorphic to the symmetric group S3.

Moreover, L´egerand Terrasson [113] proved that the outer automorphism group of the category of nonoriented maps is S3, that is, they proved that the quotient of the group of invertible elements from the category of maps onto itself, modulo the normal subgroup induced by inner automorphisms of T (∞, ∞, 2)

51 R. Nedela: Group Actions on Surfaces

is S3. As it was noted by Jones (personal communication), the above result is nothing but a restatement of Theorem 3.31 in the language of categories. Further generalisation of the above result can be found in [3]. The above results on operations on maps generalize to hypermaps as follows. Given a hypermap H = (F ; r0, r1, r2) we can derive virtually six hypermaps by permuting the three colours 0, 1, 2 of its cells; in fact, for each permutation σ ∈ S3 = S{0,1,2} we deﬁne the σ-dual DσH to be the hypermap H = (F ; rσ0, rσ1, rσ2). As expected this six operations form a group (see Machi [129]). L. James [89] proved that the outer automorphism group of T (∞, ∞, ∞) is isomorphic to P GL(2, Z). This inﬁnite group induces a set of functors generated by σ-duals and one other twisting operator.

Figure 3.15: Totally self-dual regular hypermap of type (4, 4, 4) of genus 2 arises as a quotient of the triangle T (4, 4, 4) by a characteristic subgroup .

Similar results can be derived for other categories of hypermaps [98]. As concerns functors between distinguished categories of hypermaps, Singerman’s list of triangle group inclusions [167] gives rise to a set of functors between different categories of hypermaps [98]. Let us note that the representation of a hypermap by the 3-valent 3-coloured map as well as the Walsh bipartite representation are examples of such functors. In the context of the correspondence between oriented maps and Riemann surfaces we would like to mention the following result.

Regular maps and the associated Riemann surfaces The monodromy group of an oriented regular map M of type (p, q) is an epi- morphic image of T +(p, q, 2). Let K be the kernel of this epimorphism. By Theorem 3.21 U/K is a Riemann surface which we denote by S(M), here K is considered to be a group of isometries of U. In [168, 169] Singerman and Syd- dall consider the problem whether the assignment M 7→ S(M) is injective, or in other words, whether the same Riemann surface can underlie diﬀerent regular maps. The late situation certainly happens for genus 0 regular maps, since the only Riemann surface of genus 0 is the Riemann sphere. As concerns genus 1, up to duality, there are two inﬁnite families of regular maps (maps of type (3, 6) and of type (4, 4)) and there are two Riemann surfaces associated with these

52 R. Nedela: Group Actions on Surfaces two families. To continue our discussion we need to deﬁne the following two functorial operations. By the truncation of M we mean the cubic map whose vertices are darts of M and two are joined by an edge if they form an angle of M, or they underlie the same edge. By the medial map we mean the map whose vertices are the middle points of edges of M, two being adjacent if the respective edges form an angle of M. Finally, let e(M) denotes the number of edges of M. Using the above deﬁned operations one can describe regular maps sharing the same Riemann surface in almost all cases.

Theorem 3.32 [168, 169] Let M and N be two orientably regular maps of genus > 1 with S(M) = S(N ) and e(M) ≤ e(N ). Then one of the following statements holds:

(a) N is the medial map of M,

(b) N is the truncation of M,

(e) N =∼ M or to its dual map.

Exponents In order to explain what an exponent of a map is, consider an oriented map M = (D; R,L). Recall that the map M is reflexible (mirror symmetric) if and only if M =∼ (D; R−1,L). This suggests the following definition: an integer ∼ e e will be called an exponent of M if M = Me = (D; R ,L). The mapping which realizes this isomorphism is an exomorphism of M. In general, the map Me is correctly defined provided e is coprime with the valency n of a map, i.e., with the least common multiple of lengths of cycles of R. If this is the case, the underlying graphs as well as the monodromy groups of both M and Me are identical. This is the most important property of the congruence operation ∗ associated with an exponent e ∈ Zn. For regular maps, the converse statement is proved in [150].

Theorem 3.33 [150] If M and M0 are two oriented regular maps with the same underlying graph of valency n and with identical monodromy groups. Then they 0 ∗ are congruent, i.e., M = Me for some e ∈ Zn. The exponents of a map M reduced modulo its valency form an Abelian group, the exponent group Ex(M) of M. Similarly, the exomorphisms of M form a group Exo(M), and this group is an extension of the automorphism group of M by the exponent group. The exponent group of a map M is naturally embedded in the multiplicative ∗ group Zn of invertible elements of the ring Zn of integers modulo n, n being the valency of M. Thus the order of Ex(M) divides φ(n)(φ is Euler’s function). In general, exponent groups are not functorial, but in [150] we investigate conditions under which exponents of a map can be transferred along a map homomorphism to another map.

53 R. Nedela: Group Actions on Surfaces

Inner exponents Among the exponents of a map M, so called inner exponents play a special role. The corresponding exomorphisms act on darts as inner automorphisms of the monodromy group. These give rise to a subgroup of the exponent group, the inner exponent group IEx(M) of M. In contrast to the general exponent groups, the inner exponents are functorial, which means that a map homomorphism induces a homomorphism between the inner exponent groups of the corresponding maps. The importance of inner exponents is emphasized by the fact that, apart from a short list of exceptions, inner exponent −1 implies that a regular map covers a map on a non-orientable surface. On the other hand, the fact that a map antipodally covers a map on a non-orientable surface forces −1 to be inner exponent. More details about antipodality and exponents can be ﬁnd in [150].

Generalised Petrie duality on bipartite maps In general the Petrie duality is not an operator for the category of oriented maps, since the topological Petrie dual of a map on an orientable surface may be a map on a non-orientable surface. However, for bipartite maps we have the following.

Proposition 3.34 [152] Let M be a bipartite oriented regular map. Then the following statements are equivalent: (1) M is reﬂexible,

(2) the oriented Petrie dual of M is regular.

For bipartite maps M = (D; R,L) the set of darts is a disjoint union D = D0 ∪ D1, where D0, D1 is formed by darts based at ‘black’, ‘white’ vertices, respectively. Consequently, R = R0R1, where Ri is the restriction of R onto −1 −1 Di, i = 0, 1. The oriented Petrie dual of M is a map M1/2 = (D; R0R1 ,L). Replacing −1 by another involutory exponent e of the bipartite map M we get an involutory operator acting on the set of bipartite maps. Properties of this switch operator are studied in [152]. The following statement is proved.

Proposition 3.35 Let M = (D; R,L) be a bipartite n-valent regular map and let e2 ≡ 1 mod n. Then the following statements are equivalent: (1) e ∈ Ex(M) is an exponent,

e e (2) M1/2 = (D,R0R1,L) is regular.

3.9 Lifting automorphisms of maps

Let π : X˜ → X be a covering of topological spaces. Assume we have a homeomorphism ψ : X → X. A question whether there is homeomorphism ψ˜ : X˜ → X˜ such that ψπ = πψ˜ is called the lifting automorphism problem. This problem is studied in [7, 22, 137, 208]. Particularly, the lifting automorphism problem for graphs is investigated in [19, 52, 21, 80, 106, 107, 138].

54 R. Nedela: Group Actions on Surfaces

We show how a fruitful concept of voltage assignment (see [68, 69, 70]), used to describe coverings of graphs, can be modiﬁed in order to describe homomorphisms between oriented regular maps. Voltage assignments can be used to build up regular maps from their regular quotients. The deﬁnitions and results presented in this section are taken from [140] (see also [73, 5, 6, 148, 149, 175, 97]).

Lifting condition Let M = (D; R,L) be an oriented map. An (oriented) angle of M is an ordered pair α = (x, y) = xy−→, where x and y are darts of M such that y ∈ {R(x),R−1(x),L(x)}. The angle α−1 = (y, x) is the inverse of α = (x, y). Let us denote by A(M) the set of all angles of M. An angle walk is a sequence α1α2 . . . αn of angles such that the initial dart of αi coincides with the terminal dart of αi−1 for i = 2, 3, . . . , n. Let G be a finite group. A voltage assignment on M valued in G is a function α : A(M) → G such that for any angle α ∈ A(M) one has α(α−1) = α−1(α). A voltage assignment extends from angles to walks in an obvious way by setting α(W ) = α(α1)α(α2) . . . α(αn). The group generated by voltages of closed walks based at a fixed dart x is called the local voltage group Gx. Since M is connected all the local groups are conjugate subgroups of G. Given a voltage assignment α on M = (D; R,L) valued in G set Dα = D×G and define permutations on Dα by −−→ Rα(x, h) = (R(x), hα(xRx)), −−→ Lα(x, h) = (L(x), hα(xLx)). One can prove that hRα,Lαi is transitive on Dα if and only if Gx = G. If this is the case the map Mα = (Dα; Rα,Lα) is a correctly defined map called the derived map. In what follows we shall always assume that a considered voltage assignment on M valued in G satisfies the condition Gx = G As one can expect, a regular homomorphism M˜ → M is equivalent to a natural projection Mα → M erasing the second coordinates (see [140]). It is important to stress that using angle voltage assignments we can handle also coverings of maps (map homomorphisms) which may not induce coverings between the respective underlying graphs (a disadvantage of the classical approach based on associating ordinary voltages to darts of the quotient map [69, 70]). Maybe a bit surprising is the fact that an arbitrary homomorphism defined on an orientably regular map is necessarily regular see Proposition 3.4 or [140]. It follows that angle voltage assignments provide a convenient tool for study of orientably regular maps. A natural question arises: Under what condition a voltage assignment α defined on an orientably regular map M determines an orientably regular derived map Mα? We say that a voltage assignment α : A(M) → G is locally invariant if for any τ ∈ Aut (M) and any closed walk W based at a dart x α(W ) = 1 ⇒ α(τW ) = 1. The above lifting condition appears in an explicit form firstly in [73], where map homomorphisms induced by graph coverings of the underlying graphs are investigated. The following characterization of homomorphisms between regular maps is proved in [140].

55 R. Nedela: Group Actions on Surfaces

Theorem 3.36 [140] Let M and M˜ be (oriented) regular maps. Then

if α is a locally invariant assignment on M, then the derived map Mα is (oriented) regular. if ϕ : M˜ → M is a homomorphism, then there exists a locally invariant voltage assignment α on M such that the natural projection Mα → M is equivalent to ϕ.

Let us note also that lifting part of Theorem 3.36 is, in a weaker form, proved in [73]. Weaker form of the above theorem can be found also in [138]. Voltage assignments on ﬂags of hypermaps and lifting conditions are studied by Surowski in [177]. By Theorem 3.36 coverings between regular maps are always regular with the group of covering transformations isomorphic to the voltage group. With some eﬀort one can show that every oriented regular map with a non-solvable monodromy group covers regularly either a regular map with a non-abelian simple group monodromy group, or it covers regularly a bipartite oriented regular map which partition stabilizer is either simple non-abelian, or a direct product of two isomorphic simple non-abelian groups (see [153]).

Abelian voltage group A global version of the invariance condition was used a long time ago in the proof that there are infinitely many 5-arc-transitive cubic graphs (see [19]). We say that a voltage assignment α : A(M) → G is invariant if for any τ ∈ Aut (M) and any walk W α(W ) = 1 ⇒ α(τW ) = 1. Of course, an invariant voltage assignment is locally invariant as well. If the assignment α : A(M) → G is invariant, then Aut (Mα) is a split extension of G by Aut (M) (see [140]). Each map M admits at least one invariant volt- n age assignment which can be constructed as follows: Let G = Z2 , where n is |A(M)|/2. Take a set X = {ε1, ε2, . . . , εn} of generators of G and let τ be a bijection A(M) → X. Then τ is an invariant voltage assignment. Since the local voltage group is not equal to G, the derived map splits into isomorphic connected parts. A restriction of the covering onto any connectivity component yields a map homomorphism such that Aut (M) lifts. In particular, if M is regular then so is the lifted map. Another example of an invariant voltage assignment can be constructed by assigning the voltage 1 ∈ (Z2, +) to each angle of M. The double cover is connected if and only if the truncation of M is not bipartite. The lifting condition significantly simplifies if the voltage group is abelian. Since a map homomorphism between regular maps with a solvable automorphism groups decomposes into a sequence of coverings, each with an elementary abelian group of covering transformations, elementary abelian covers of maps such that the map automorphism group lifts are of special importance. Voltage assignments in elementary abelian groups were studied by Malniˇc, Maruˇsiˇcand Potoˇcnikin [141] and by Du, Kwak and Xu in [54]. Using some ideas of linear algebra an algorithm to check the lifting condition is derived. Invariant voltage assignments defined on graphs and valued in products of cyclic

56 R. Nedela: Group Actions on Surfaces groups are used in [19, 21, 175, 138] as well. Regular cyclic covers over Platonic solids are described in [176] and [97], regular cyclic covers over toroidal regular maps are classiﬁed in [166]. Homology and cohomology theories of hypermaps are built in works of Machi [130], Surowski and Schroeder [178]. The following construction shows that every regular map of genus g admits a locally invariant ‘homology voltage assignment’ of rank 2g.

Theorem 3.37 (Theorem 7.1. in [178]) Let M be a regular orientable map of Euler characteristic 2 − 2g and genus g. Then for each positive integer n, there is a regular (connected) map of Euler characteristic 2gn(2 − 2g) covering M.

3.10 Regular embeddings of graphs Cayley maps A Cayley map is an oriented map M, which underlying graph is a Cayley graph C(Γ,X) such that the local rotation of generators at any vertex is induced by a cyclic permutation ρ of the generating set X. The above condition implies that the colour group Γ ∈ Aut (M) acts as a group of map automorphisms. Sur- prisingly large number of significant constructions in topological graph theory (explicitly or implicitly) include a construction of a Cayley maps. The notion of Cayley maps was first used by Biggs in [18] as a name for a class of regular Cay- ley maps called nowadays balanced Cayley maps. Cayley complete maps were used to solve the Heawood map coloring problem [156]. As we can see below all regular embeddings of complete graphs, complete bipartite graphs and of hypercubes are Cayley maps. A systematic study of regular Cayley maps started by works of Siráˇnandˇ Skovieraˇ [174, 171] and continued in articles of Jajcay, Con- der, Tucker, Richter and others (see for instance [20, 82, 83, 84, 85, 155, 86, 193]). An exhaustive information on Cayley maps is collected in [155]. A sample of results related to regular Cayley maps follow.

Theorem 3.38 [155] Let M = (D; R,L) be a regular map of valency r. Then M is a Cayley map if and only if there is a homomorphism ϕ : Mon(M) → Sym(r) sending R 7→ (1, 2, . . . , r).

The above theorem suggests that regular Cayley maps of the same valency r split into ﬁnitely many classes, Cayley maps belonging to the same class cover the same embedding of a one-vertex graph. Particularly, if the one-vertex map is regular then the Cayley map M is balanced, i.e., the distribution ρ of generators satisﬁes (ρx)−1 = ρ(x−1) for all x ∈ X. In this particular case, the covering is regular and it induces a locally invariant voltage assignment in Γ. The local invariance is in this case equivalent to the statement: “the local rotation of generators ρ of Γ extends to a group automorphism of Γ”. The latter condition was known already for Biggs and White ( see [18, 191]). Regularity and other related properties of balanced Cayley maps are studied in [174]. For regular balanced Cayley map of valency n the monodromy group (or equivalently) the map automorphism group is a split extension Zn : K of the cyclic group of order n by the colour group K acting regularly on vertices of the map. In general, the monodromy group hR,Li of a regular Cayley map is

57 R. Nedela: Group Actions on Surfaces a product hRi · K. More precisely, we have the following statement proved by Jajcay and Sir´aˇn[86].ˇ

Theorem 3.39 Let M be an n-valent regular map with monodromy group G = hR,Li. Then M is a Cayley map if and only if there exists a subgroup K of G such that hRi · K = G and K ∩ hRi = 1G.

Skew-morphisms of groups and regular Cayley maps The monodromy group of an n-valent balanced regular Cayley map based on a group K can be costructed as an external semidirect product. We have

Proposition 3.40 [174] Let M be a Cayley map based on a group K deﬁned by a cyclic permutation of the generators ρ. Then M is a regular balanced Cayley map if and only if ρ extends to an automorphism of K.

Theorem 3.39 and Proposition 3.40 suggest that the following generalisation of a group automorphism could be useful to study the structure of monodromy groups (automorphism groups) of Cayley regular maps. Let H be a ﬁnite group, ϕ : H −→ H a permutation of H of order k (in the full symmetric group Sym(H)), and π : H −→ Zk a function from H into the cyclic group Zk. We say that ϕ is a skew-morphism of H, with an associated power function π, if ϕ(1H ) = 1H and

ϕ(ab) = ϕ(a)ϕπ(a)(b) for all a, b ∈ H where ϕπ(a)(b) is the image of b under ϕ applied π(a) times. Proposition 3.40 generalizes as follows.

Theorem 3.41 [86] Let M be a Cayley map based on a group K = hx1, x2, . . . , xni deﬁned by a cyclic permutation of generators ρ = (x1, x2, . . . , xn). Then M is regular if and only if ρ extends to a skew-morphism of K.

The following two propositions summarise known properties of skew morphisms (see Jajcay-Sir´aˇn[86]).)ˇ

Proposition 3.42 [86] Let ϕ be a skew-morphism of a ﬁnite group H and let π be the power function of ϕ. Then the following hold: (a) The set ker π = {a ∈ H | π(a) = 1} is a subgroup of H,

(b) π(g) = π(h) if and only if g and h belong to the same right coset of the subgroup ker π in H, (c) The set F ixϕ = {a ∈ H | ϕ(a) = a} is a subgroup of H, (d) π(ghg−1) = 1 for all h ∈ ker π ∩ F ixϕ and all g ∈ H

(e) The group ker π ∩ F ixϕ is a normal subgroup of F ixϕ.

Skew-morphism of abelian groups satisfy some additional properties (Conder- Jajcay-Tucker [46, 47]).

58 R. Nedela: Group Actions on Surfaces

Proposition 3.43 If A is a ﬁnite abelian group, and ϕ is a skew-morphism of A, then (i) ϕ preserves ker π set-wise (that is, ϕ(ker π) = ker π), (ii) the restriction of ϕ to ker π is a group automorphism of ker π, and (iii) for each a in A, the power π(a) is congruent to 1 modulo the length of every non-trivial orbit of ϕ on ker π. Given skew-morphism σ over an abelian group A consider the decomposition O of A into orbits. The following hold: (i) {0} ∈ O, (ii) if o ∈ O then the set o−1 = {−x|x ∈ o} ∈ O is an orbit.

It follows that orbits of σ form a system of basic sets over Schur ring over Zn. This observation allows as to apply the theory of Schur rings to investigate skew-morphisms over abelian groups. In [47] the following problem is investigated in detail.

Problem. Which abelian groups admit regular Cayley maps?

Since the multiplication by −1 extends to a group automorphism of a cyclic group, each cyclic group (Zn, +) admits a trivial balanced regular Cayley map by taking the generating set to be {1, −1}. As concerns the existence of non-trivial balanced regular Cayley maps based on cyclic groups the following statement was proved in [47].

Theorem 3.44 [47] For odd m > 1 the cyclic group Zm has a non-trivial balanced regular Cayley map if and only if m is not a product of distinct Fermat primes one of which is 3.

Skew-morphisms with kernel of index 1 are group automorphisms. Skew- morphism with kernel of index 2 are called t-automorphisms in [126]. If a group G admits a skew-morphism with kernel K of index 2 then G is a Z2-extension of K. In fact, all t-automorphisms can be described in terms Z2-extensions of the kernel K and its automorphism group Aut (K) (see [126] for details).

Classification of regular embeddings of given graph The classification of regular maps by the underlying graph was initiated by Heffter [78] who constructed regular maps by embedding the complete graph Kp, p a prime, into the surface of genus (p − 1)(p − 4)/4 or (p2 − 7p + 4)/4 according to whether p ≡ 1 or p ≡ 3 (mod 4). The classification of regular embeddings of the complete graph has been completed by work of several authors, see [16, 88] and [87, 200]. In [150] the following approach to classification of (orientably) regular maps with a given underlying graph divided into three stages is suggested. The first stage consists in characterizing all the finite graphs that underlie some regular map. At the theoretical level, this stage is completed: necessary and sufficient conditions for a graph G to underlie a regular map (oriented

59 R. Nedela: Group Actions on Surfaces or nonoriented, reflexible or irreflexible) have been given by Gardiner, Nedela, Siráˇnandˇ Skovieraˇ [62]. The conditions are expressed in terms of the automorphism group of a graph. In the case of oriented maps the existence of a group G ≤ Aut (G) such that the action on darts of G is regular with cyclic stabilizer of a vertex is required. For the nonoriented case the statement follows.

Theorem 3.45 [62] A connected graph G of valency at least 3 is the underlying graph of a regular map if and only if its automorphism group contains a subgroup G such that

(1) G acts transitively on the set of darts of G,

(2) the edge stabilizer Ge of an edge is dihedral of order 4,

(3) the stabilizer Gv of a vertex v is dihedral with a cyclic group of index two acting regularly on darts incident to v.

At the second stage, one has to analyze the automorphism group of a graph G satisfying the above mentioned necessary and sufficient conditions and to specify the appropriate subgroups in Aut (G) to be the map automorphism groups. Of course, it is sufficient to consider one representative of every conjugacy class. Finally, at the third stage, one is dealing with one of the subgroups of Aut (G) chosen at the previous stage, say G. One then determines all the regular maps whose automorphism group is G. As shown in Section 3.8, this step in the classification process can be successfully accomplished. Indeed, if (G; R) and (G; R0) are two regular maps (with the same underlying graph G) whose automorphism group is G, then there is an integer e coprime with the valency of G, say n, such that R0 = Re (Theorem 3.33). It follows that the isomorphism classes of maps M with Aut (M) = G correspond to the cosets of the exponent group ∗ ∗ Ex(G,R) in Zn, and their number is |Zn : Ex(M)|. I what follows by a regular embedding we mean an embedding of a graph such that the corresponding map is regular in the category of MAPS. An orientably regular embedding will be an embedding of a graph into an orientable surface such that the corresponding map is orientably regular.

Complete maps The graph-theoretical approach to the classification problem can be nicely il- lustrated by the example of complete graphs. A map determined by a 2-cell embedding of a complete graph into a closed surface will be called a complete map. It follows from the general characterization [62] that an automorphism group G of an orientably regular embedding of the complete graph on n vertices Kn acts regularly on the set of darts of Kn with cyclic vertex stabilizer of order n − 1. Since the set of darts of Kn can be identified with all pairs (a, b) ∈ Zn × Zn, the action of G on the set of vertices is sharply transitive. Such groups are well-known, it follows that n = pk is a power of prime and ∼ k G ≤ Sn = Aut (Kn) is the group of affine transformations of GF (p ). It was proved by James and Jones [88] that every oriented regular map of Kn is isomorphic to the map Mt = (D; R,L) for some primitive element t of the field, where D = Zn × Zn, L(a, b) = (b, a) and R(a, b) = (a, a + (b − a)t). More- over, two such maps are isomorphic if and only if the corresponding primitive

60 R. Nedela: Group Actions on Surfaces elements are conjugate under the Galois group Gal(pk) (see [150]). The latter is a consequence of the fact that the exponent group Ex(Mt) is isomorphic in k k φ(p −1) a very natural way with Gal(p ). Thus there are k nonisomorphic orientably regular embeddings of Kn, where φ(n) is the Euler function. Hence we have Theorem 3.46 The following statements hold:

k (1) An orientably regular embedding of Kn (n ≥ 2) exists if and only if n = p is a power of prime; (2) A complete map M = (D; R,L) with n = pk vertices is regular if and only ∼ k if M = Mt for some primitive element t of GF (n) = GF (p ).

k k φ(p −1) (3) Let n = p is a power of prime. There are k orientably regular embeddings of Kn.

Note that the above maps Mt are balanced Cayley maps with the distribu- ∗ tion of generators in Zn determined by the multiplication by t. Of course this multiplication determines an automorphism of Zn and hence by Proposition 3.40 all the maps Mt are orientably regular. It is a bit surprising that not all complete graphs admit oriented regular embeddings. As concerns regular embeddings the condition on n is even more restrictive. It is proved in [87, 200] that Kn admits such an embedding if and only if n is 2,3,4 or 6. Thus the maps Mt form a family of chiral maps provided n = pk ≥ 5.

Complete bipartite maps

It is well-known that for each integer n, the complete bipartite graph Kn,n has at least one regular embedding in an orientable surface described by Biggs and White [20, §5.6.7] as a Cayley map for the group Z2n. In [154] it is shown that there is exactly one regular embedding of Kp,p, p a prime. This result was generalised in [100] proving that there is a unique regular embedding of Kn,n if and only if square of a prime does not divide n and p - q − 1 for any two prime divisors of n. Such prime divisors of n are called disjoint. Note that the set of n admitting a unique regular embedding of Kn,n coincides with the set of n such that there is a unique group of order n. Perhaps, the most general result is proved in [101], where the regular embeddings of Kn,n are classiﬁed in case when n is odd and any two prime divisors p, q of n are disjoint. Q Theorem 3.47 For n = k pei , where p are distinct primes i = 1, . . . , k, Q i=1 i i k ei−1 denote ν(n) = i=1 pi = n/p1 . . . pk. If n is odd then the number of oriented regular embeddings of Kn,n is ν(n) ≥ ρ(n), with equality if and only if the primes dividing n are mutually disjoint. e−1 In particular, there are exactly p oriented regular embeddings of Kpe,pe , p an odd prime.

e−1 e All the p non-isomorphic regular embeddings of Kn,n (n = p a power of an odd prime) are of type {2n, n} and of genus (n − 1)(n − 2)/2. The method of the proof uses the following fact: if M is an oriented regular map with the + underlying graph Kn,n for some n then the subgroup G = Aut0 M < Aut (M)

61 R. Nedela: Group Actions on Surfaces of automorphisms of M, preserving vertex-colours, factorises as a product of two disjoint cyclic groups of order n. Generally, the structure of groups which are products of two cyclic groups is not known. The main obstacle seems to be a lack of information in the case when the considered product is not metacyclic. When n is an odd prime power, a result of Huppert [81] implies that such a group G must be metacyclic, and this enabled us to classify the possibilities for G, and hence for M. When p = 2, however, Huppert’s result does not apply, e and indeed for each e ≥ 2 there are regular embeddings of Kn,n (n = 2 ) for which G is not metacyclic, so these are not direct analogues of the maps arising when p is odd. Nevertheless, the techniques used in the odd case can also be applied here, giving a partial classiﬁcation for p = 2, provided one restricts to those embeddings for which G is metacyclic. Recently, the regular embeddings ∼ ∼ of complete bipartite graphs with G = HK, where H = K = Z2e are isomorphic cyclic 2-groups, were classiﬁed [57]. It turned out that given e > 2 there are exactly 4 regular maps with non-metacyclic G < Aut (M). Hence we have

Theorem 3.48 Let n = 2e be a power of two, e > 2. Then there are exactly e−2 2 + 4 oriented regular embeddings of Kn,n.

For n = 2 there is one, and for n = 4 there are two regular embeddings of Kn,n, respectively.

Example 3.49 A family of regular embeddings of K2n,2n, n ≥ 2 with a non- metacyclic colour preserving subgroup G can be obtained from the regular toroidal map M = {4, 4}n,n ﬁrst taking the Petrie dual P(M) and then applying the duality operator D. Since M is a reﬂexible regular bipartite map, the map D(P(M)) is regular. Since M has two parallel sets of n Petrie polygons of length n, each polygon having no edges in common with the other polygons in its own set, and one edge in common with each polygon in the other set (see Fig. 3.16); it follows that the vertices and edges of On = D(P(M)) have the same incidence properties, that is, the underlying graph of the map is K2n,2n. Since its faces are 4-gons, On is a minimum-genus embedding of K2n,2n. To prove that G is not metacyclic one needs to analyse its structure, in particular, its derived group [57].

Figure 3.16: Petrie polygons in {4, 4}4,4.

62 R. Nedela: Group Actions on Surfaces

The classiﬁcation of regular embeddings of complete bipartite graphs is still not completed. Some new results were proved by using an approach in [108] based on the following theorem.

Theorem 3.50 Let M be an oriented regular embedding of Kn,n. Then there 0 0 0 is an assignment of vertices V → {0, 1, . . . , n − 1} ∪ {0 , 1 ,..., (n − 1) } of Kn,n such that the vertex stabiliser at a vertex assigned by 0 is hρi ≤ Aut (M), where 0 0 0 ρ = σ(0 , 1 ,..., (n − 1) ) and σ is a skew-morphism of Zn of order d|n with a −x power function π(x) = −σ (−1). Moreover, each such skew-morphism of Zn determines exactly one regular embedding of Kn,n.

Using the above theorem Kwak and Kwon [109] proved that for n odd there is exactly one (reﬂexible) regular embedding of Kn,n, namely the standard one ∼ constructed by Biggs and White with G = Zn × Zn and Aut (M) = (Zn × Zn): Z2.

Regular embeddings of n-cubes

The n-dimensional cube Qn is a graph whose vertex-set V is formed by the vectors of dimension n over the field GF (2), two vertices are adjacent if they differ in precisely one coordinate. The existence of at least two regular embeddings of Qn for every n was known a long time ago. For instance, Q3 has the well-known 4-gonal regular embedding in the sphere, as well as a hexagonal regular embedding in the torus, the map {6, 3}2,0 in the Coxeter-Moser notation [51, page 107]. In [149] the following family of regular maps with the underlying graph Qn is constructed. Let {e0, e1,..., en−1} denote the standard basis for V. Then, an arc of Qn is an ordered pair of vertices (v, v + ej) for some v ∈ V and for some j. The dart-reversing involution L is given by 2 L(v, v+ej) = (v+ej, v). Let e be an element of Zn such that e ≡ 1 ( mod n). Set R(v, v + ej) = (v, v + ej+e||v|| ), where ||v|| is the sum of components of v modulo 2. Denote by M(n, e) = (Qn; R) = (D; R,L) the above map. In [149] is conjectured that there are no other oriented regular embeddings of Qn. In [58] the conjecture is confirmed for odd n.

Theorem 3.51 Let Qn be the n-dimensional cube, where n is odd. Then any oriented regular map M with the underlying graph Qn is isomorphic to M(n, e) for some integer e satisfying e2 ≡ 1 ( mod n). Moreover, diﬀerent solutions of the congruence e2 ≡ 1 ( mod n) give rise to non-isomorphic maps.

In contrast, in [110] a lot of regular embeddings of Qn not belonging in the above family are constructed for every even n, n ≥ 6. Thus the conjecture fails for even integers n ≥ 6. The classiﬁcation problem for even n remains open. Moreover, there are no regular embeddings of Qn into non-orientable surfaces for n > 2, [111].

Other families of graphs Regular embeddings of several other families of were considered. The classiﬁ- cation is known for the tensor products Kn ⊗ K2 [148, 149]; for the complete multipartite graphs Kn[K¯p], p a prime, [56]; for graphs with p and pq vertices, p, q primes, [55]; for merged Johnson’s graphs in [99]. Regular embeddings of

63 R. Nedela: Group Actions on Surfaces graphs with multiple edges are considered in [115]. For further results on regular embeddings of graphs see also [116, 117].

Lifting of classification Sometimes, we are able to “lift” the classification of regular embeddings of a graph K to the classification of regular embeddings of K˜ , where K˜ covers K. This approach is used in [148, 149] to classify oriented regular embeddings of the tensor product Kn ⊗ K2 and of all oriented regular maps with two faces. The maps with two faces were previously dealt with by Brahana [24], Coxeter and Moser [51], Vince [185] and Garbe [61]. Surprisingly, the latter classification result can be employed in deriving a simple arithmetic condition for a generalized Petersen graph to be a Cayley graph [147]. The method of lifting regular maps wrapped by exponents from the base graph to its canonical double covering can be generalized to other coverings. In a more general approach developed in [149], the wrapping is controlled by a homomorphism from the group of covering transformations to the exponent group of the base map which is to be lifted with wraps. The technique of wrapped lifts produces many new regular maps and seems to be very fruitful and promising for the construction and classification of bipartite regular maps. In [152] the above construction is explained from the point of view of the switch operator introduced in Section 3.8. Particularly, using this technique the above mentioned oriented regular embeddings of n-dimensional cubes were constructed in [149].

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74 Chapter 4

Minimal triangulations of given edge width

Triangulations of the sphere, and more generally, of closed surfaces are maximal 2-cell embeddings of simple graphs. Because of that many difficult problems on graph embeddings, including the four colour problem and its generalisations, reduce to problems on triangulations. In what follows we shall discuss some generalisations of the well-known statement of Steinitz establishing that every polyhedral triangulation arises from the tetrahedron by a finite sequence of vertex-splittings. The following text is in part based on Section 5.4. of monograph of Mohar and Thomassen [19]. Let T be a triangulation of a surface. Let e be an edge of T which lies in exactly two 3-cycles of the underlying graph of T . These 3-cycles are facial. We can get a new triangulation T 0 by contracting the edge e and replacing the resulting two pairs of double edges by simple ones. We say that e can be contracted and that the triangulation T 0 is obtained from T by edge-contraction, vice-versa, we say that T can be obtained from T 0 by vertex-splitting. The triangulation T of is said to be minimal if no edge of T can be contracted. For surfaces other than the sphere minimal triangulations T are characterized by the following property: each edge of T lies in a non- contractible 3-cycle. Generally, given k ≥ 3 a triangulation T of a surface is said to be k-minimal if each edge of T lies in a non-contractible k-cycle. The only minimal triangulation of the sphere is K3 (and it is the tetrahedron when we require the triangulations to be 3-connected [28]). Hence every spherical triangulation can be obtained from K3 using the vertex-splitting operation. Similarly, the basis for the set of polyhedral triangulations is formed by the tetrahedron, and every polyhedral triangulation arises from the tetrahedron applying a sequence of vertex-splittings. Vertex splittings preserving the minimum degree of a triangulations are considered by Nakamoto and Negami [22]. Barnette [2] proved that there are exactly two minimal triangulations for the projective plane. Lawrencenko [14] described the minimal triangulations for the torus (there are 21 of them). Lawrecenko and Negami [15] determined the 25 minimal triangulations for the Klein bottle. Barnette and Edelson [3, 4] proved that the set of minimal triangulations is finite for every fixed compact surface. A simple proof of this fact was discovered by Gao, Richmond and Thomassen [8]. The proof of Nakamoto and Ota [21] gives, moreover, an asymptotically

75 R. Nedela: Group Actions on Surfaces linear upper bound O(g) on the size of minimal triangulations of genus g. The five 4-minimal triangulations for the projective plane were determined by Fisk, Mohar and Nedela [7]. A fundamental result establishing the finiteness of the set of k-minimal triangulations on a fixed surface for any fixed k was proved by Malniˇcand Nedela [17] (see the attached paper).

Theorem 4.1 (Malniˇcand Nedela [17]) Let S be a surface of genus g and let k ≥ 3 be an integer. There is a constant ck,g such that every k-minimal triangulation on S has at most ck,g edges.

Crucial argument in our proof consists in proving a bound on the number of homotopy classes of cycles passing through a given vertex. Later Gao, Richter and Seymour [9] gave a diﬀerent and shorter proof of Theorem 4.1. Yet another proof was found by Juvan, Malniˇcand Mohar in [13] (see [[19],pages 144-146] as well). The proof in [9] gives a bound ck2k!(4kk!)kg2 on the number of vertices on a k-minimal triangulation on Sg (for some constant c). Some more related results on k-minimal triangulations can be found in Seress and Szabo [27]; Clark, Entringer, McCanna and Sz´ekekly [6]; Hutchinson [11, 12] and Przytycka, Przytycki [25]. Let i : K 7→ S be a 2-cell embedding of a (multi)-graph K into a surface S. The face-width of i is the minimum number of faces of i whose closure contains cycle which is non-contractible in S. We say that i is minimal of face-width k ≥ 2 if it has face width k but for any edge e the edge contraction or edge deletion yields an embedding of face width less than k. The following theorem follows from Theorem 4.1 (see [17], [19, page 154]).

Theorem 4.2 (Malniˇcand Nedela [17]) For every (compact) surface S and every integer k ≥ 2 there are only ﬁnitely many minimal embeddings of face- width k in S.

Taking into the account that edge-width and face-with of a non-spherical triangulation coincide, we see that Theorem 4.2 is in fact a generalization of Theorem 4.1. Alternatively, Theorem 4.2 can be proved using the Robertson- Seymour theory of minors using the following deep result. A partial order of a set is said to be well-quasi order if it does not give an inﬁnite anti-chain. We say that a graph G is a minor of H and write G << H if G arises from H by a sequence of vertex deletions and edge contractions.

Theorem 4.3 (Robertson and Seymour [29, page 334]) The set of ﬁnite graphs is well-quasi ordered with respect <<.

Embeddings of graphs of face-width at least k (for some k ≥ 3), in particular triangular embeddings, share a lot of interesting combinatorial and topological properties which are extensively discussed in Chapter 5 of [19] and in [24] where one can ﬁnd more references. A sample of relevant references is attached.

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78 Chapter 5

Publication record and citation index

5.1 Publication record of the author Scientiﬁc papers in books: 1. Nedela,R.-Skoviera,M.:ˇ On graphs embeddable with short faces, in “Top- ics in Combinatorics and Graph Theory” (Bodendiek, R. Henn Eds.), Physica-Verlag Heidelberg, 1990, 519–529.

2. Nedela,R.-Skoviera,M.:ˇ Atoms of cyclic connectivity in transitive cubic graphs, in “Contemporary Methods in Graph Theory” (R. Bodendiek ed.), Mannheim BI - Wissenschafts - Verlag, 1990, 479–488.

Scientiﬁc papers in journals included in the ISI-web of science: 1. Nedela,R.-Skoviera,M.:ˇ The maximum genus of vertex-transitive graphs, Discrete Math. 78, 1989, 179–186.

2. Nedela,R.-Skoviera,M.:ˇ The maximum genus of a graph and doubly Eu- lerian Trails, Bolletino U.M.I. (7) 4-B, 1990, 541–551. 3. Nedela,R.: Locally homogeneous graphs with dense links at vertices, Cze- choslovak Math. J. 42 (117), 1992, 515–517. 4. Nedela,R.: Covering spaces of locally homogeneous graphs, Discrete Math. 121, 1993, 177–188. 5. Fisk,S.-Mohar,B.-Nedela,R.: Minimal locally cyclic triangulations for the Projective plane, J. Graph Theory 18 (1), 1994, 25–35. 6. Nedela,R.: Covering projections of graphs preserving links of vertices and edges, Discrete Math. 134, 1994, 111–124.

7. Nedela,R.-Skoviera,M.:ˇ Which generalized Petersen graphs are Cayley graphs?, J. Graph Theory 19 (1), 1995, 1–11.

79 R. Nedela: Group Actions on Surfaces

8. Nedela,R.-Skoviera,M.:ˇ Regular embeddings of canonical double coverings of graphs, J. Combin. Theory B 67 (2), 1996, 249–277. 9. Nedela,R.-Skoviera,M.:ˇ Decompositions and reductions of snarks, J. Graph Theory 22 (3), 1996, 253–279.

10. Nedela,R.-Skoviera,M.:ˇ Exponents of orientable maps, Proceedings of Lon- don Math. Soc. (3) 75, 1997, 1–31.

11. Nedela,R.-Skoviera,M.:ˇ Regular maps from voltage assignments and exponent groups, European J. Combin. 18, 1997, 807–823.

12. Maruˇsiˇc,D.-Nedela,R.:Maps and half-transitive graphs of valency 4, Eu- ropean J. Combin. 19, 1998, 345–354.

13. Gardiner,A.-Nedela,R.-Sir´aˇn,J.-ˇ Skoviera,M:ˇ Characterization of graphs which underlie regular maps on closed surfaces. J. London Math. Soc. 59 (1), 1999, 100–108.

14. Grasselli,L.-Mulazzani,M.-Nedela,R.: 2-Symmetric Transformations for 3- Manifolds of Genus 2, J. Combin. Theory B 79, 2000, 105–130. 15. Malniˇc,A.-Nedela,R.-Skoviera,M.:ˇ Lifting graph automorphisms by voltage assignments, European J. Combin. 21, 2000, 927–947.

16. Marusiˇc,D.-Nedela,R.: On the point stabilizers of transitive groups with non-self-paired suborbits of length 2, J. Group Theory 4, 2001, 19–43. 17. Nedela,R.-Skoviera,M.:ˇ Regular maps on surfaces with large planar width, European J. Combin. 22, 2001, 243–261. 18. Nedela,R.-Skoviera,M.:ˇ Cayley snarks and almost simple groups, Combi- natorica 21, 2001, 583–590. 19. Malniˇc,A.-Nedela,R.-Skoviera,M.:ˇ Regular homomorphisms and regular maps, European J. Combin. 23 (4), 2002, 449–461. 20. Nedela,R.-Skoviera,M.-Zlatoˇs,A.:Regularˇ embeddings of complete bipartite graphs, Discrete Math. 258 (1-3), 2002, 379–381.

21. Marusiˇc,D.-Nedela,R.: Finite graphs of valency 4 and girth 4 admitting half-transitive group actions, J. Austral. Math. Soc. 73, 2002, 155–170.

22. Du,S.F.-Kwak,J.H.-Nedela,R.: Regular maps with pq vertices, J. Alge- braic Combin. 19, 2004, 123–141. 23. Breda,A.-Nedela,R.: Half-arc-transitive graphs and regular hypermaps, European J. Combin. 25, 2004, 423–436. 24. Hor´ak,P.-Nedela,R.-Rosa,A.: The Hamilton-Waterloo Problem: the case of Hamilton Cycles and Triangle-Factors, Discrete Math. 284, 2004, 181– 188.

25. Du,S.F.-Kwak,J.H.-Nedela,R.: Regular embeddings of complete multipartite graphs, European J. Combin. 26, 2005, 505–519.

80 R. Nedela: Group Actions on Surfaces

26. Breda,A.-Nedela,R.-Sir´aˇn,J.:Classiﬁcationˇ of regular maps of prime negative Euler characteristic, Transactions of Amer. Math. Soc. 357, 2005, 4175-4190. 27. Kwon,Y.S.-Nedela,R.: Non-existence of regular embeddings of n-dimensional cubes, accepted in Discrete Math.

Scientific papers in journals currented by Math. Reviews (but not by the ISI-web of Science): a) foreign journals 1. Haviar,A.-Nedela,R.: On Varieties of Graphs, Discussiones Mathematicae, Graph Theory 18, 1998, 209–223. 2. Nedela,R.-Skoviera,M.-Zlatoˇs,A.:ˇ Bipartite maps, Petrie duality and exponent groups, Collection of papers published in honour of M. Pezzana, edited by C. Galliardi, Atti Sem. Mat. Fis. Univ. Modena, Suppl. 49, 2001, 109–133. 3. Nedela,R.: Regular maps - combinatorial objects relating different fields of mathematics. Mathematics in the new millennium (Seoul, 2000). J. Korean Math. Soc. 38 (5), 2001, 1069–1105. 4. Breda,A.-Nedela,R.: Chiral hypermaps of small genus, Contributions to Algebra and Geometry 44 (1), 2003, 127–143. 5. Breda,A.-Breda,A.-Nedela,R.: Chirality group and chirality index of Cox- eter toroidal maps, Ars Comb., in print. b) domestic journals 1. Nedela,R.-Skoviera,M.:ˇ Remarks on diagonalizable embeddings of graphs, Math. Slovaca 40, 1990, 15–20. 2. Nedela,R.: On a problem from extremal graph theory, Acta Math. Univ. Com. LVI-LVII, Alfa, Bratislava, 1990, 11–18. 3. Nedela,R.-H´ıc,P.-Pavliková,S.:Front-Divisors of Trees, Acta Math. Univ. Com. LXI (1), 1992, 69–84. 4. Nedela,R.: Locally cyclic graphs covering the complete tripartite graphs, Math. Slovaca 42, 1992, 143–146. 5. Nedela,R.: Edge-locally homogeneous graphs, Acta Univ. M. Belii, Ser. Math. 1, 1993, 27–32. 6. Nedela,R.-Malniˇc,A.: K-minimal triangulations of surfaces, Acta Math. Univ. Com. LXIV (1), 1995, 57–77. 7. Nedela,R.-H´ıc,P.: Note on zeros of the characteristic polynomial of balanced trees, Acta Univ. Math. Belli, Ser. Math. 3, 1995, 31–36. 8. Nedela,R.-Skoviera,M.:ˇ Atoms of cyclic connectivity in cubic graphs, Math. Slovaca 45, 1995, 481–499.

81 R. Nedela: Group Actions on Surfaces

9. Nedela,R.-Skoviera,M.-Jendrol’,S.:ˇ Constructing regular maps form their planar quotients. Math. Slovaca 47, 1997, 155–170.

10. Nedela,R.: Graphs which are edge-locally Cn, Math. Slovaca 47, 1997, 381–391.

11. H´ıc,P.-Nedela,R.: Balanced integral trees, Math. Slovaca 48, 1998, 429– 445.

12. Marusiˇc,D.-Nedela,R.:Partial line graph operator and half-arc-transitive group actions, Math. Slovaca 51 (3), 2001, 241–257.

13. Breda,A.-Nedela,R.: Join and intersection of hypermaps, Acta Univ. Math. Belii, Ser Math. 9, 2001, 13–28.

14. Breda,A.-Nedela,R.: Chiral hypermaps with few hyperfaces, Math. Slo- vaca 53, 2003, 107–108.

15. Karab´aˇs,J.-Nedela,R.:Minimal representatives of G-classes of 3-manifolds of genus two, Acta Univ. Math. Belii, Ser. Math. 10, 2003, 21–45. 16. Meszka,M.-Nedela,R.-Rosa,A.: Circulants and the chromatic index of Steiner triple systems, accepted Math. Slovaca.

Scientiﬁc papers in conference proceedings: 1. Nedela,R.-Dakic,T.-Pisanski,T.: Embeddings of Tensor Product Graphs, Graph Theory and Algorithms (Proceedings of the seventh quadriennial international conference on the theory and applications of graphs), Wiley- Interscience pub., New York, 1995, 893–905. 2. Nedela,R.: Regular maps and Groups acting Half - transitively on Graphs of valency 4. Northern Arizona University, Flagstaﬀ, Arizona, July 20–24, 1998, 20–22.

3. Nedela,R.-Weissensteiner,J.-Zimkov´a,M.:What is the rigth exchange rate of the Slovak crown to other currencies? Acta Oeconomica Pragensia 8, 2000 ˇc.3,95–98.

4. Nedela,R.-Nedelová,G.: Efficient portfolio problem - analysis of Slovak capital market, Proceedings of research seminar Mathematics, Statististics and Economy, Faculty of Economics Matej Bel University, 1998, 29–37. 5. Breda,A.-Nedela,R-Siráˇn,J:Classificationˇ of regular maps of prime negative Euler characteristic, Proceedings of Com2Mac Mini-Workshop on Hurwitz Theory and Ramifications, Pohang University of Science and Technology, Korea 2003, 89–99.

6. Nedela,R.: Regular maps-combinatorial objects relating different fields of mathematics, Proceedings of Com2Mac Mini-Workshop on Hurwitz The- ory and Ramifications, Pohang University of Science and Technology, Ko- rea 2003, 333–361.

82 R. Nedela: Group Actions on Surfaces

7. Du,S.F.-Kwak,J.H.-Nedela,R.: A classiﬁcation of regular embeddings of graphs of order a product of two primes, Proceedings of Com2Mac Mini- Workshop on Hurwitz Theory and Ramiﬁcations, Pohang University of Science and Technology, Korea 2003, 333–361.

8. Breda,A.-Jones,G.-Nedela,R.-Skoviera,M.:ˇ Chirality group and chirality index of regular maps and hypermaps, Proceedings of Com2MaC Mini- Workshop on Two-face embeddings of graphs and applications, POSTECH Pohang 2004, Eds. M.M. Deza and J.H.Kwak.

Textbooks: 1. Haviar,A.-Hrnˇciar,P.-Nedela,R.-Snoha,L’.: “Zbierka úlohstredoslovenského koreˇspondenˇcnéhosemináraz matematiky”, KP BanskáBystrica, 1990.

2. Nedela,R.: “Prij´ımacieskúˇskyz matematiky na Fakulte financi´ıv rokoch 1998-2000, Fakulta financi´ı UMB”, BanskáBystrica, 2001. 125 strán, Náklad1000 ks.

5.2 Citation index of the author

The record is organised as follows: a work of the author typeset in boldface is followed by a list of works of other authors in which the above work is cited. Each particular list is organised chronologically. Items included in the database of ISI-web of science are assigned by [SCI]. Citations in monographs are identiﬁed by [M]. A partial summary of the form x + y + z is included, the ﬁrst number x denotes the number of citations of the considered work of the author according to the ISI-web of Science, the second number y denotes the number of citations in monographs, the third number z denotes the number of other collected citations. Citations in a work with a non-empty intersection of set of authors with the set of authors of the cited work are excluded. Nedela,R.-Skoviera,M.:ˇ The maximum genus of vertex-transitive graphs, Discrete Mathematics 78, 1989, 179–186. Cited in: 4+2+1 1. McCuaig,W.D.: Edge reductions in cyclically k-connected cubic graphs, J. Combin. Theory B 56, 1992, 16–44. [SCI] 2. Huang,Y.Q.-Liu,Y.P.: On the upper embeddability of graphs, Science in China (Ser. A) 28, 1998, 223–228. (Chinese) [SCI] 3. Huang,Y.Q.-Liu,Y.P.: Upper embeddability of graphs, Science in China (Ser. A) 41, 1998, 498–504. [SCI] 4. Huang,Y.Q.-Liu,Y.P.: Face size and the maximum genus of a graph I. Simple graphs, J. Combin. Theory B 80, 2000, 356–370. [SCI] 5. Huang,Y.Q.-Liu,Y.P.: Face size and the maximum genus of a graph II. Multigraphs, Math. Slovaca 51, 2001, 129–140.

83 R. Nedela: Group Actions on Surfaces

6. Chen,J.: Minimum and maximum embeddings, in Gross,J.L.-Yellen,J.: “Handbook of graph theory”, CRC Press, Boca Raton, 2004, 625–641. [M] 7. Gross,J.L.-Tucker,T.W.: “Topological graph theory”, Dover Publications, New York, 2001. [M]

Nedela,R.: On a problem from extremal graph theory, Acta Univ. Math. Com. LVI-LVII, Alfa, Bratislava, 1990, 11–18. Cited in: 0+0+2 1. Markov´a,I.:Decompositions of complete graphs into factors with diameter two, Acta Math. Univ. Com., 1990, 235–243. 2. H´ıc,P.-Palumb´ıny,D.: Isomorphic factorization of complete graphs into factors with given diameter, Math. Slovaca 37, 1997, 247–253.

Nedela,R.-Skoviera,M.:ˇ On graphs embeddable with short faces, in Topics in Combinatorics and Graph Theory (Bodendiek, Henn Eds.), Physica-Verlag Heidelberg, 1990, 519–529. Cited in: 6+0+5 1. Nebesk´y,L.:Local properties and upper embeddability of connected multigraphs. Czechoslovak Math. J. 43 (118) (2), 1993, 241–248. [SCI] 2. Nebesk´y,L.: Certain cubic multigraphs and their upper embeddability, Czechoslovak Math. J., 45 (120), 1995, 385–392. [SCI] 3. Huang,Y.Q.-Liu,Y.P.: Some classes of upper embeddable graphs, Acta Math. Scientia 17, 1997, (suppl.), 154–161. 4. Huang,Y.Q.-Liu,Y.P.: Degree-sum of non-adjacent vertices and upper- embeddability of graphs, Chinese Ann. Math. Ser. A 19, 1998, 651–656. 5. Huang,Y.Q.-Liu,Y.P.: On the upper embeddability of graphs, Science in China (Ser. A) 28, 1998, 223–228. (Chinese) [SCI] 6. Huang.Y.Q.-Liu Y.P.: On the upper embeddability of planar-embeddable graphs, J. Systems Sci. Mat. Sci. 19, 1999, 415–419. 7. Huang.Y.Q.-Liu Y.P. Maximum genus and girth of a graph, J. Math. Research Exp. 20, 2000, 187–193. 8. Huang,Y.Q.-Liu,Y.P.: Face Size and the Maximum Genus of a Graph 1. Simple Graphs, Journal of Combin. Theory B 80 (2), 2000, 356–370. [SCI] 9. Huang,Y.Q.-Liu Y.P.: Face size and the maximum genus of a graph 2. Nonsimple graphs. Math. Slovaca 51, 2001, 356–370. 10. Huang,Y.Q.: Maximum genus of a graph in terms of its embedding properties Discrete Math. 262 (1-3), 2003, 171–180. [SCI] 11. Huang,Y.Q.: Maximum genus and chromatic number of graphs, Discrete Math. 271 (2003), 117–127. [SCI]

84 R. Nedela: Group Actions on Surfaces

Nedela,R.-Skoviera,M.:ˇ The maximum genus of a graph and doubly Eulerian Trails, Bolletino U.M.I. (7) 4-B, 1990, 541–551. Cited in: 2+1+6 1. Fleischner,H.: “Eulerian graphs and related topics”, Part 1, Vol. 2, Annals of Discrete Math. 50, North Holland, 1991, (page VIII.6 and A.22). [M] 2. Fu,H.L.-Tsai,M.Ch.: Edge and vertex operations on upper embeddable graphs, Math. Slovaca 46, 1996, 9–19. 3. Fu,H.L.-Tsai,M.C.: On the upper embeddability of diameter three graphs, Australas. J. Comb. 14, 1996, 187–197. 4. Huang,Y.Q.-Liu,Y.P.: Some classes of upper embeddable graphs, Acta Math. Scientia 17, 1997, (suppl.), 154–161. 5. Huang,Y.Q.-Liu,Y.P.: Extensions on 2-edge connected 3-regular up-embeddable graphs, Acta Math. Appl. Sinica 14, 1998, 337–346. 6. Huang,Y.Q.-Liu,Y.P.: Degree-sum of non-adjacent vertices and upper- embeddability of graphs, Chinese Ann. Math. Ser. A 19, 1998, 651–656. 7. Huang,Y.Q.-Liu,Y.P.: An improvement of a theorem on the maximum genus for graphs, Math. Applicata 11, 1998, 109–112. (Chinese) 8. Huang,Y.Q.: Maximum genus of a graph in terms of its embedding properties Discrete Math. 262 (1-3), 2003, 171–180. [SCI] 9. Huang, Y.Q.: Maximum genus and chromatic number of graphs, Discrete Math. 271, 2003, 117–127. [SCI]

Nedela,R.-Skoviera,M.:ˇ Atoms of cyclic connectivity in transitive cubic graphs, in Contemporary Methods in Graph Theory (R. Bo- dendiek ed.), Mannheim BI - Wissenschafts - Verlag, 1990, 479–488. Cited in: 4+1+0 1. McCuaig,W.D.: Edge reductions in cyclically k-connected cubic graphs, J. Combin. Theory B 56, 1992, 16–44. [SCI] 2. Kochol,M.: Snarks without small cycles, J. Combin. Theory B 67, 1996, 34–47. [SCI] 3. Zhang,C.Q.: “Integer ﬂows and cycle covers of graphs”, Marcel Dekker, New York, 1997. [M] 4. Cavicchioli,A.-Meschiari,M.-Ruini,B. et al.: A survey on snarks and new results: Products, reducibility and a computer search, J. Graph Theory 28 (2), 1998, 57–86. [SCI] 5. VanDenHeuvel,J.-Jackson,B.: On the edge-connectivity, hamiltonicity, and toughness of vertex-transitive graphs, J. Combin. Theory B 77, 1999, 138– 149. [SCI]

Nedela,R.-Skoviera,M.:ˇ Remarks on diagonalizable embeddings of graphs, Math. Slovaca 40, 1990, 15–20. Cited in: 0+0+2

85 R. Nedela: Group Actions on Surfaces

1. Demoviˇc,A.: A note on the hamiltonian genus of a complete bipartite graph, Acta Math. Univ. Comenian. 64, 1995, 77–81. 2. Abu-Sbeih,M.: Diagonalizable embeddings of composition graphs, Math. Slovaca 49, 1999, 479–488.

Nedela,R.: Locally homogeneous graphs with dense links at vertices, Czechoslovak Math. J. 42, (117), 1992, 515–517. Cited in: 0+0+1

1. Solt´es,L.:Edgeˇ neighbourhoods in line graphs, Acta Math. Univ. Come- nian. 62, 1993, 161– 167.

H´ıc,P.-Nedela,R.-Pavl´ıková,S.: Front-divisors of trees, Acta Math. Univ. Comeniane, vol. LXI, 1, 1992, 69–84. Cited in: 0+0+1 1. Pokorný,M.-Sotáková,K.:On nonsymetric strong integral digraphs, CO- MAT-TECH 2000, 8.medzinárodnákonferencia, Materiálovotechnologická fakulta STU, Trnava, 19. - 20. októbra2000, 173–178.

Nedela,R.: Covering spaces of locally homogeneous graphs, Discrete Math. 121, 1993, 177–188. Cited in: 0+0+1

1. Orlovich,Yu.L.: Coverings by cliques, factors and graphs with isomorphic vertex neighborhoods, Diskretn. Anal. Issled. Oper Ser. 19 no.2, 2002, 48–90. (Russian)

Fisk,R.-Mohar,B.-Nedela,R.: Minimal locally cyclic triangulations for projective plane, J. Graph Theory 18 (1), 1994, 25–35. Cited in: 0+2+1

1. Liebers: Planarazing graphs - A survey and Annotated bibliography, J. Graph Algorithms Appl. 5, 2001, no. 1, 74 pages (electronic). 2. Gross,J.L.-Tucker,T.W., “Topological Graph Theory”, Dover Publications, New York, 2001. [M] 3. Negami,S.: Triangulations, in Gross,J.L.-Yellen,J.: “Handbook of Graph Theory”, [M] CRC Press, Boca Raton, 2004, 737–760.

Nedela,R.: Covering projections of graphs preserving links of vertices and edges, Discrete Math. 134, 1994, 111–124. Cited in: 0+0+2

1. Solt´es,L.:Edgeˇ neighbourhoods in line graphs, Acta Math. Univ. Comen. 62, 1993, 161–167.

2. Fronˇcek,D.: Grafy s konstantn´ım okol´ım vrcholu a hran, Kandidátska dizertaˇcnápráca,MFF UK v Bratislave, 1992.

86 R. Nedela: Group Actions on Surfaces

Nedela,R.-Dakiˇc,T.-Pisanski,T.:Embeddings of tensor product graphs, “Graph Theory and Algorithms” (Proceedings of the seventh qua- drenial international conference on theory and applications of graphs), Wiley - Int., New York, 1995, 893–905. Cited in: 1+1+0

1. Abay-Asmerom,G.: On genus imbeddings of the tensor products of graphs, J. Graph Theory 23 (1), 1996, 67–76. [SCI]

2. Gross,J.L.-Tucker,T.W., “Topological Graph Theory”, Dover Publications, New York, 2001. [M]

Nedela,R.-Malniˇc,A.:K-minimal triangulations of surfaces, Acta Math. Univ. Com. LXIV (1), 1995, 57–77. Cited in: 7+1+3

1. Archdeacon,D.: “Topological Graph Theory”; A survey, http://www.emba.uvm.edu/~archdeacon, 1995. 2. Gao,Z.-Richter,B.-Seymour,P.: Irreducible triangulations of surfaces, J. Combin. Theory B 68, 1996, 206–217. [SCI]

3. Brunet,R.-Nakamoto,A.-Negami,S.: Diagonal ﬂips of triangulations preserving speciﬁed properties, J. Combin. Theory B 68, 1996, 295–309. [SCI] 4. Mohar,B.: Face-width of embedded graphs, Mathematica Slovaca 47 (1), 1997, 35– 64.

5. Randby,S.P.: Minimal embeddings in the projective plane, J. Graph The- ory 25 (2), 1997, 151–163. [SCI] 6. Lawrecenko,S.-Negami,S.: Constructing the graphs that triangulate both torus and the Klein Bottle, J. Combin. Theory B 77, 1999, 211–218. [SCI]

7. Mohar,B.-Thomassen,C.: Graphs and surfaces, J. Hopkins Univ. Press, Baltimore, 2001. [M] 8. Mohar,B.-Robertson,N.: Flexibility of Polyhedral Embeddings of Graphs in Surfaces, J. Combin. Theory B 83, 2001, 38–57. [SCI] 9. Negami,S.: Crossing numbers of graph embedding pairs on closed surfaces J. Graph Theory 36 (1), 2001, 8–23. [SCI] 10. Negami,S: Triangulations, in Gross,J.L.-Yellen,J.: “Handbook of Graph Theory”, CRC Press, Boca Raton, 2004, 737–760. [M] 11. Geelen, J.F.- Richter, R.B.- Salazar, G.: Embedding grids in surfaces, European J. Combin. 25 (6) , 2004, 785–792. [SCI]

Nedela,R.-Skoviera,M.:ˇ Which generalized Petersen graphs are Cay- ley graphs?, J.Graph Theory 19 (1), 1995, 1–11. Cited in: 5+1+9

87 R. Nedela: Group Actions on Surfaces

1. Sir´aˇn,J.:“Graphˇ embeddings”, DrSc. Thesis, Comenius Univ., Bratislava, 1993. 2. Saraˇzin,M.L.-Pacco,W.: I graﬁ di Petersen generalizzati, Preprint ser. Di di Matematica, Politecnico di Milano, n. 208/P.

3. Marauta,E.A.: Generalized Petersen graphs of genus 1, J. Comb. Inf. Syst. Sci. 19, 1994, 25–30.

4. Jendrol’,S.-Zold´ak,V.:ˇ The irregularity strength of generalized Petersen graphs, Math. Slovaca 45, 1995, 107–113.

5. Jajcay,R.-Sir´aˇn,J.:ˇ More constructions of vertex transitive graphs, non- Cayley graphs, based on counting closed walks, Australas J. Combin. 14, 1996, 121–132.

6. Saraˇzin,M.L.: A note on the generalized Petersen graphs that are also Cayley graphs, J. Combin. Theory B 69, 1997, 226–229. [SCI]

7. Scapellato,R.: Vertex-transitive graphs and digraphs, in “Graph Sym- metry, Algebraic methods and applications”, edited by G. Hahn and G. Sabidussi, Kluwer Academic Publ., London, 1997. 8. Cowan,D.-Lin,J.: Which generalized Petersen graphs are Cayley graphs - a short proof, Univ. of Letterbridge, Preprint. 9. Maruˇsiˇc,D.-Pisanski,T.:The remarkable generalized Petersen graph G(8,3), Math. Slovaca 50, 2000, 117–121. 10. Baˇca,M.:Consecutive-magic labeling of generalized Petersen graphs, Uti- litas Mathematica 58, 2000, 237–241. [SCI] 11. Feng,Y.Q-Kwak,J.H.: “Cubic s-regular graphs”, COM2MAC Lecture Notes (7), 2000. [M]

12. Alspach,B.-Qin Y.S.: Hamilton-connected Cayley graphs on Hamiltonian groups, European J. Combin. 22 (6), 2001, 777–787. [SCI] 13. Malniˇc,A.:Action graphs and coverings, Discrete Math. 244 (1-3), 2002, 299–322. [SCI]

14. Feng,Y.Q.: On vertex-transitive graphs of odd prime-power order, Dis- crete Math. 248 (1-3), 2002, 265–269. [SCI] 15. Feng,Y.Q.-Kwak,J.H.: Cubic symmetric graphs of order a small number times a prime or a prime square, Com2MaC Preprint Ser. 2003, No. 2.

Nedela,R.-Skoviera,M.:ˇ Atoms of cyclic connectivity in cubic graphs, Math. Slovaca 45, 1995, 481–499. Cited in: 2+0+0 1. Cavicchioli,A.-Meschiari,M.-Ruini,B.-Spaggiari,F.: A survey on snarks and new results: Products, Reducibility and Computer search, J. Graph The- ory 28 (2), 1998, 57–86. [SCI]

88 R. Nedela: Group Actions on Surfaces

2. Dvoˇrák,Z.-Kára,J.-Král,D.-Pangrác,O.:An Algorithm for Cyclic Edge Con- nectivity of Cubic Graphs, in Algorithm Theory - SWAT 2004: 9th Scan- dinavian Workshop on Algorithm Theory, Humlebaek, Proceedings Eds: T. Hagerup, J. Katajainen, Lecture Notes in Computer Science, Springer- Verlag Heidelberg Vol. 3111, 2004, 236–247. [SCI]

Nedela,R.-Skoviera,M.:ˇ Regular embeddings of canonical double coverings of graphs, J. Combin. Theory B 67 (2), 1996, 249–277. Cited in: 6+1+9

1. Siráˇn,J.:Mapyˇ a pr´ıbuznékombinatorickéˇstruktúry, Doktorskádizertaˇcná práca(DrSc.), MFF UK Bratislava, 1993.

2. Gvozdiak,P.-Sir´aˇn,J.:Regularˇ maps from voltage assignments, in Contem- porary Mathematics 147: Graph Structure Theory (N. Robertson and P. Seymour, Eds.), Providence, 1993, 441–454.

3. Scapellato,R.: Vertex-transitive graphs and digraphs, in Graph Symmetry, Algebraic methods and applications, edited by G. Hahn and G. Sabidussi, Kluwer Academic Publ., London, 1997, 319–378.

4. Pacco,W.-Scapellato,R.: Generalized orbital graphs, Preprint ser. Di di Matematica, Politecnico di Milano n. 225/P, 1996. 5. Pacco,W.-Scappellato,R.: Digraphs having the same canonical double covering, Discrete Math. 173, 1997, 291–296. [SCI] 6. Surowski,D.B.: Stability of arc-transitive graphs, J. Graph Theory 38 (2), 2001, 95–110. [SCI] 7. Gross,J.L.-Tucker,T.W.: “Topological Graph Theory”, Dover Publica- tions, New York, 2001. [M]

8. Wilson,S.: Families of regular graphs in regular maps, J. Combin. Theory B 85 (2) 2002, 269–289. [SCI] 9. Anderson,K.-Surowski,D.B.: Coxeter-Petrie Complexes of Regular Maps European J. Combin. 23, 2002, 861–880. [SCI] 10. Jones,G.: Graphs, groups and surfaces, Rendiconti del Seminario di Mat. Messina, Ser. II, Suppl. No. 8, 2002, 71–85.

11. Kwak,J.H.: Tesselations and regular maps, Commun. Korean Math. Soc. 18, 2003, 1–20. (Korean). 12. Pacco,W.-Scapellato,R.: K-coverins of digraphs, Preprint Politecnico di Milano. 13. Surowski,D.: Automorphisms of certain unstable graphs, Math. Slovaca 53, 2003, 215–232.

14. Kwak,J.H.-Kwon,Y.S.: Exponent and switch exponent groups of regular orientable embeddings of complete bipartite graphs, CoM2MaC Preprint Ser., 2003, No. 4.

89 R. Nedela: Group Actions on Surfaces

15. Kwon,Y.S.: New regular embeddings of n-cubes Qn, J. Graph Theory 46, 2004, 297–312. [SCI] 16. Jones,G.A.: Automorphisms and regular embeddings of merged Johnson graphs, European J. Combin. 26, 2005, 417–435. [SCI]

Nedela,R.-Skoviera,M.:ˇ Decompositions and reductions of snarks, J. Graph Theory 21 (4), 1996, 253–279. Cited in: 7+0+6 1. Steffen,E.: On bicritical snarks, Preprint 97-027, SFB 343, Univ. Bielefeld, 1997. 2. Cavicchioli,A.-Meschiari,M.-Ruini,B.-Spaggiari,F.: A survey on snarks and new results: Products, Reducibility and Computer search, J. Graph The- ory 28 (2), 1998, 57–86. [SCI] 3. Steffen,E.: Classifications and characterizations of snarks, Discrete Math. 188, 1998, 183–203. [SCI] 4. Pisanski,T.: Kubiˇcnigrafi, Obzornik mat. fiz. 45, 1998. (Slovenian) 5. Brinkmann,G.-Steffen,E.: Snarks and reducubility, Ars Combinatoria 50, 1998, 292–296. [SCI] 6. Steffen,E.: Non-bicritical critical snarks, Graphs and Combin. 15, 1999, 473–480. [SCI] 7. Grunewald,S.-Steffen,E.: Cyclically 5-edge connected non-bicritical critical snarks, Discuss. Math. Graph Theory 19, 1999, 5–11. 8. Steffen,E: On bicritical graphs, Math. Slovaca 51, 2001, 253–279. 9. Cavicchioli,A.-Murgolo,T.E.-Ruini,B.-Spaggiari,F.: Special classes of snarks, Acta Appl. Math. 76, 2003, 57–88. [SCI] 10. Potoˇcnik,P.: Edge-colourings of cubic graphs admitting a solvable vertex- transitive group of automorphisms, Univ. Ljubljana, Preprint Ser. 39, 2001, No. 781. 11. Hrnˇciar,P.: On color-closed multipoles, Acta Univ. Math. Belii, Ser. Math. 7, 1999, 31–34. 12. Potoˇcnik,P:Edge-colourings of cubic graphs admitting a solvable vertex- transitive group of automorphisms, J. Combin. Theory B 91 (2), 2004, 289–300. [SCI] 13. Dvoˇrák,Z.-Kára,J.-Král,D.-Pangrác,O.:An Algorithm for Cyclic Edge Con- nectivity of Cubic Graphs, in Algorithm Theory - SWAT 2004: 9th Scan- dinavian Workshop on Algorithm Theory, Humlebaek, Proceedings Eds: T. Hagerup, J. Katajainen, Lecture Notes in Computer Science, Springer- Verlag Heidelberg Vol. 3111, 2004, 236–247. [SCI]

H´ıc,P.-Nedela, R.: A note on characteristic polynomials of integral trees, Acta Univ. Mathaei Belii, Ser. Math., No. 3, 1996, 31–35. Cited in: 0+0+2

90 R. Nedela: Group Actions on Surfaces

1. Tóth,P.: O reláciách a ich grafoch, Acta Fac. Paed. Univ. Tyrnaviensae, Ser. B, 1998, No. 2, 165–173. 2. Pokorný,M.-Sotáková,K.:On nonsymetric strong integral digraphs, CO- MAT-TECH 2000, 8.medzinárodnákonferencia, Materiálovotechnologická fakulta STU, Trnava, 19. - 20. 10. 2000, 173–178.

Nedela,R.-Skoviera,M.:ˇ Exponents of orientable maps, Proceedings of London Math. Soc., (3) 75m, 1997, 1–31 Cited in: 4+3+5

1. Sir´aˇn,J.:“Graphˇ embeddings”, DrSc. Thesis, Comenius Univ., Bratislava, 1993.

2. Archdeacon,D.-Gvozdjak,P.-Sir´an,J.:ˇ Constructing and forbidding automorphisms in lifted maps, Mathematica Slovaca 47 (2), 1997, 113–129.

3. Liu,Y.P: “Enumerative theory of maps”, Kluwer Acad. Pub., Dordrecht, 1999. [M]

4. Gross,J.L.-Tucker,T.W.: “Topological Graph Theory”, Dover Publica- tions, New York, 2001. [M]

5. Feng,Y.Q.-Kwak,J.H.: Cubic s-regular graphs, COM2MAC Lecture Notes (7), 2002. [M] 6. Malniˇc,A.:Action graphs and coverings, Discrete Math. 244 (1-3), 2002, 299–322. [SCI] 7. Feng,Y.Q.-Kwak,J.H.: Constructing an inﬁnite family of cubic 1-regular graphs, European J. Combin. 23, 2002, 559–565. [SCI] 8. Wang,K.S.-Kwak,J.H.: Frobenius maps, CoM2MaC, Preprint Ser. 2002, No. 21. 9. Kwak,J.H.-Kwon,Y.S.: Exponent and switch exponent groups of regular orientable embeddings of complete bipartite graphs, CoM2MaC Preprint Ser., 2003, No. 4.

10. Kwak,J.H.: Tesselations and regular maps, Commun. Korean Math. Soc. 18, 2003, 1–20. (Korean)

11. Kwon,Y.S.: New regular embeddings of n-cubes Qn, J. Graph Theory 46, 2004, 297–312. [SCI] 12. Li,C.H.-Sir´aˇn,J.:ˇ Regular maps whose groups do not act faithfully on vertices, edges, or faces, European J. Combin. 26, 2005, 521–541. [SCI]

Nedela,R.-Skoviera,M.:ˇ Regular maps from voltage assignments and exponent groups. European J. Combin. 18, 1997, 807–823. Cited in: 5+3+3

1. Malniˇc,A.:Group actions, coverings and lifts of automorphisms, Discrete Math. 182 (1-3), 1998, 203–218. [SCI]

91 R. Nedela: Group Actions on Surfaces

2. Sir´aˇn,J.:Theˇ ‘walk calculus’ of regular lifts of graph and map automorphisms, Yokohama Math. J. 47, 1999, 113–128. 3. Feng,Y.Q.-Kwak,J.H.: Cubic s-regular graphs, CoM2MaC Lecture Notes (7), 2000. [M] 4. Kwak,J.H.-Kwon,Y.S.: Generalized Cayley graphs, CoM2MaC, Preprint Ser. , 2000, No.2. 5. Gross,J.L.-Tucker,T.W.: “Topological Graph Theory”, Dover Publica- tions, New York, 2001. [M] 6. Feng,Y.Q.-Kwak,J.H.: Constructing an inﬁnite family of cubic 1-regular graphs, European J. Combin. 23, 2002, 559–565. [SCI] 7. Feng,Y.Q.-Wang,K.S.: s-regular cyclic coverings of the three-dimensional hypercube Q(3), European. J. Combin. 24 (6), 2003, 719–731. [SCI] 8. Kwak,J.H.-Kwon,Y.S.: Exponent and switch exponent groups of regular orientable embeddings of complete bipartite graphs, CoM2MaC, Preprint Ser. 2003, No. 9. 9. Feng,Y.Q.-Kwak,J.H.: s-regular cubic graphs as coverings of the complete bipartite graph K3,3, J. Graph Theory 45 (2), 2004, 101–112. [SCI] 10. Gross,J.L.: Voltage graphs, in Gross,J.L.-Yellen,J.: “Handbook of Graph Theory”, CRC Press, Boca Raton, 2004, 661–683. [M]

11. Kwon,Y.S.: New regular embeddings of n-cubes Qn, J. Graph Theory 46, 2004, 297–312. [SCI]

Jendrol’,S.-Nedela,R.-Skoviera,M.:ˇ Constructing regular maps from their planar quotients, Math. Slovaca 47, 1997, 155–170. Cited in: 7+2+3 1. Kochol,M.: Snarks without small cycles, J. Combin. Theory B 67, 1996, 34–37. [SCI] 2. Archdeacon,D.-Gvozdjak,P.-Siráˇn,J.:Constructingˇ and forbidding automorphisms in lifted maps, Mathematica Slovaca 47 (2), 1997, 113–129. 3. Voss,H.J.: Sachs triangulations, generated by dessins denfants, and regular maps, Mathematica Slovaca 47 (2), 1997, 193–210. 4. Cavicchioli,A.-Meschiari,M.-Ruini,B.-Spaggiari,F.: A survey on snarks and new results: Products, Reducibility and Computer search, J. Graph The- ory 28 (2), 1998, 57–86. [SCI] 5. Voss,H.-J.: Sachs triangulations and infinite sequences of given type, Dis- crete Math. 191, 1998, 223–240. [SCI] 6. Siráˇn,J.:Triangleˇ group representations and constructions of regular maps, Proceedings of London Math. Soc. 82 (3), 2001, 1–34. [SCI] 7. Siráˇn,J.:Triangleˇ group representations and their applications to graphs and maps Discrete Math. 229 (1-3), 2001, 341–358 . [SCI]

92 R. Nedela: Group Actions on Surfaces

8. Gross,J.L.-Tucker,T.W.: “Topological Graph Theory”, Dover Publica- tions, New York, 2001. [M] 9. Sir´aˇn,J.:“Regularˇ maps”, CoM2MaC Lecture Notes No. 2, POSTECH, Pohang, 2001. [M]

10. Cavicchioli,A.-Murgolo,T.E.-Ruini,B.-Spaggiari,F.: Special classes of snarks, Acta Appl. Math. 76 (1), 2003, 57–88. [SCI]

11. Li,C.H.-Sir´aˇn,J.,Regularˇ maps whose groups do not act faithfully on vertices, edges, or faces, European J. Combin. 26, 2005, 521–541. [SCI]

12. R.B. Richter, J.Sir´aˇn,R.ˇ Jajcay, T.W. Tucker and M.E. Watkins, Cayley maps, preprint.

Nedela,R.: Graphs which are edge-locally Cn, Math. Slovaca, 47, No. 4, 1997, 381–391. Cited in: 4+0+0 1. Fronˇcek,D.: Graphs with given edge neighborhood, Czechoslovak Math. J. 39, 1989, 627–630. [SCI] 2. FronˇcekD.: Graphs with near v-neighborhoods and e-neighborhoods, Glasgow Math. J., 32,1990, 197–199. [SCI]

3. FronˇcekD.: Edge-realizable graphs with universal vertices, Glasgow Math. J., 33,1991 309–310. [SCI]

4. Solt´es,.:Regularˇ graphs with regular neighbourhoods, Glasgow Math. J. 34, 1992, 215–218. [SCI]

Nedela,R.-Maruˇsiˇc,D.:Maps and half-transitive graphs of valency 4. European J. Combin., 19, 1998, 345–354. Cited in: 12+1+0

1. Li,C.H.-Sim,S.: On half-transitive metacirculant graphs of prime-power order, J. Combin. Theory B 81, 2001, 45–57. [SCI] 2. Feng,Y.Q.-Kwak,J.H.: Constructing an inﬁnite family of cubic 1-regular graphs, European J. Combin. 23, 2002, 559–565. [SCI] 3. Feng,Y.Q-Kwak,J.H.: Cubic s-regular graphs, COM2MAC Lecture Notes (7), 2002. [M] 4. Feng,Y.Q-Kwak,J.H.: An inﬁnite family of cubic one-regular graphs with unsolvable automorphism groups, Discrete Math. 269 (1-3), 2003, 281– 286. [SCI]

5. Feng,Y.Q-Wang,K.S.: s-regular cyclic coverings of the three-dimensional hypercube Q(3), European J. Combin. 24 (6), 2003, 719–731. [SCI] 6. Kwak,J.H.-Oh,J.M.: Inﬁnitely many ﬁnite one-regular graphs of any even valency, J. Combin. Theory B 90 (1), 2004, 185–191. [SCI]

93 R. Nedela: Group Actions on Surfaces

7. Oh,J.M.-Hwang,K.W.: Construction of one-regular graphs of valency 4 and 6, Discrete Math. 278 (1-3), 2004, 195–207. [SCI] 8. Feng,Y.Q.-Kwak,J.H.: s-regular dihedral coverings of the complete graph of order 4, Chinese Ann. Math. B 25 (1), 2004, 57–64. [SCI]

9. Feng,Y.Q.-Kwak,J.H.: s-regular cubic graphs as coverings of the complete bipartite graph K3,3 J. Graph Theory 45 (2), 2004, 101–112. [SCI] 10. Wilson, S.: Semi-transitive graphs J. Graph Theory 45 (1), 2004, 1–27. [SCI]

11. Feng,Y.Q.-Kwak,J.H.-Wang,K.: Classifying cubic symmetric graphs of order 8p and 8p2, European J. Combin., (In press). [SCI]

12. Fang,X.G.-Li,C.H.-Xu,M.Y.:On edge-transitive Cayley graphs of valency four, European J. Combin. 25, 2004, 1107–1116. [SCI]

13. Feng,Y.Q.-Kwak,J.H.: One-regular cubic graphs of order a small number times a prime or a prime square, J. Australian Math. Society, 76, 2004, 345–356. [SCI]

H´ıc,P.-Nedela,R.: Balanced integral trees, Math. Slovaca, 48, No. 5, 1998, 429–445. Cited in: 3+0+5

1. Zámoˇz´ık,J.:Uber Computer in der Ingenieurgeometrie, Geometrie - Tagung, TU Graz, 1999, 218–227. 2. Pokorný,M.- Sotáková,K.:On nonsymetric strong integral digraphs, CO- MAT-TECH 2000, 8.medzinárodnákonferencia, Materiálovotechnologická fakulta STU, Trnava, 19. - 20. 10. 2000, 173–178.

3. Baliˇnska,K.T.-Simiˇc,S.K.:The nonregular, bipartite, integral graphs with maximum degree four, part I: Basic properties, Discrete Math. 236, 2001, 13–24. [SCI] 4. Pokorn´y,M.:Integral balanced trees and Pells equations, Acta Fac. Paed. Univ. Tyrnaviensis, Ser. C, 2002, 44–50. 5. Wang,L.-Li,X.-Hoede,C.-Xueliang,Li.: Integral complete r-partite graphs, Discrete Math. 283 (1-3), 2004, 231–241. [SCI] 6. Wang,L.-Li,X.-Zhang,S.: Families of integral trees with diameters 4, 6, and 8, Discr. Appl. Math. 136 (2-3), 2004, 349–362. [SCI] 7. Baliˇnska K., CvetkoviˇcD.-RadosavljeviˇcZ.-SimiˇcS.-StevanoviˇcD. : A Survey on Integral Graphs, Univ. Beograd. Publ. Elektrotechn.Fak. Ser. Mat 13, 2002, 42-65.

8. Pokorn´y,M.: A Note on Integral Balanced Rooted Trees of Diameter 10. Acta Fac. Paed. Univ. Tyrnaviensis, Ser. C, 2004, no. 8. (in print)

94 R. Nedela: Group Actions on Surfaces

Gardiner,A.-Nedela,R.-Sir´aˇn,J.-ˇ Skoviera,M.:ˇ Characterisation of graphs which underlie regular maps on closed surfaces, J. London Math. Soc. 59 (1999), 100–108. Cited in: 6+1+4

1. Jajcay,R.: Characterization and construction of Cayley graphs admitting regular Cayley maps, Discrete Math. 158, 1996, 151–160. [SCI]

2. Gross,J.L.-Tucker,T.W.: “Topological Graph Theory”, Dover Publica- tions, New York, 2001. [M]

3. Wilson,S.: Families of regular graphs in regular maps, J. Combin. Theory B 85 (2), 2002, 269–289. [SCI]

4. Malniˇc,A.:Action graphs and coverings, Discrete Math. 244 (1-3), 2002, 299–322. [SCI] 5. Kwak,J.H.: Tesselations and regular maps, Commun. Korean Math. Soc. 18, 2003, 1–20. (Korean)

6. Surowski,D.: Automorphism groups of certain unstable graphs, Math. Slovaca 53, 2003, 215–232. 7. Kwak,J.H.-Kwon,Y.S.: Generalized Cayley graphs, Com2MaC Preprint Ser., 2003, No.1. 8. Kwak,J.H.-Kwon,Y.S.: Exponent and switch exponent groups of regular embeddings of complete bipartite graphs, Com2MaC Preprint Ser., 2003, No.4.

9. Kwak,J.H.-Oh,J.M.: Inﬁnitely many ﬁnite one-regular graphs of any even valency, J. Combin. Theory B 90, 2004, 185–191. [SCI]

10. Kwon,Y.S.: New regular embeddings of n-cubes Qn, J. Graph Theory 46, 2004, 297–312. [SCI]

11. Feng, Y.Q.-Kwak, J.H.-Wang,K.:Classifying cubic symmetric graphs of order 8p or 8p2, European J. Combin. (In press) [SCI]

Malniˇc,A.-Nedela,R.-Skoviera,M.:ˇ Lifting graph automorphisms by voltage assignments, European J. Combin. 21, 2000, 927–947. Cited in: 10+3+6 1. Potoˇcnik,P.: Edge-colourings of cubic graphs admitting a solvable vertex- transitive group of automorphisms, Univ. Ljubljana, Preprint Ser. 39, 2001, No. 781.

2. Surowski,D.: Stability and arc-transitivity of graphs, J. Graph Theory, 38, 2001, 95–110. [SCI] 3. Sir´aˇn,J.:“Regularˇ maps”. CoM2MaC Lecture Notes (2), 2001. [M]

4. Sir´aˇn,J.:Coveringsˇ of graphs and maps, orthogonality, and eigenvectors J. Algebr. Combin. 14 (1), 2002, 57–72. [SCI]

95 R. Nedela: Group Actions on Surfaces

5. Feng,Y.Q.-Kwak,J.H.: Constructing an inﬁnite family of cubic 1-regular graphs, European J. Combin. 23, 2002, 559–565. [SCI] 6. Feng,Y.Q.-Kwak,J.H.: Cubic s-regular graphs, COM2MAC Lecture Notes (7), 2002. [M] 7. Feng,Y.Q.-Kwak,J.H.: s-Regular dihedral coverings of the complete graph of order 4, CoM2MaC Preprint Ser., 2002, No. 18. 8. Kwak,J.H.-Oh,J.H.: One-regular graphs of any even valency with non- cyclic vertex stabilizer, CoM2MaC Preprint Ser., 2002, No. 18. 9. Feng,Y.Q.-Kwak,J.H.: One regular cubic graphs of order a small number times a prime or a prime square, CoM2MaC Preprint Ser., 2002, No. 19. 10. Maruˇsiˇc,D.:On 2-arc-transitivity of Cayley graphs, J. Combin. Theory B 87, 2003, 162–196 . [SCI] 11. Feng,Y.Q.-Wang,K.S.: s-regular cyclic coverings of the three-dimensional hypercube Q(3), European J. Combin. 24 (6), 2003, 719–731. [SCI] 12. Feng,Y.Q.-Kwak,J.H.: Cubic symmetric graphs of order a small number times a prime or a prime square, Com2MaC Preprint Ser. 2003, No. 2. 13. Feng,Y.Q.-Kwak,J.H.-Wang,K.: Classifying cubic symmetric graphs of order 8p or 8p2, CoM2MaC Preprint Ser., 2003, No. 8. 14. Du,S.F.-Kwak,J.H.-Xu,M.Y.: Linear criteria for lifting automorphisms of elementary abelian regular coverings, Linear Algebra Appl. 373, 2003, 101–119. [SCI] 15. Feng,Y.Q.-Kwak,J.H.: s-Regular cubic graphs as coverings of the complete bipartite graph K3,3, J. Graph Theory 45 (2), 2004, 101–112. [SCI] 16. Feng,Y.Q.-Kwak,J.H.: s-regular dihedral coverings of the complete graph of order 4, Chinese Ann. Math. B 25 (1), 2004, 57–64. [SCI] 17. Gross,J.L.: Voltage graphs, in Gross,J.L.-Yellen,J.: “Handbook of Graph Theory”, CRC Press, Boca Raton, 2004, 661–683. [M] 18. Potoˇcnik,P:Edge-colourings of cubic graphs admitting a solvable vertex- transitive group of automorphisms, J. Combin. Theory B 91 (2), 2004, 289–300. [SCI] 19. Feng,Y.Q.-Kwak,J.H.-Wang,K.: Classifying cubic symmetric graphs of order 8p and 8p2, European J. Combin., In press. [SCI]

Nedela,R.-Skoviera,M.:ˇ Regular maps on surfaces with large planar width, European J. Combin. 22 (2), 2001, 243–261. Cited in: 4+2+0 1. Maruˇsiˇc,D.:Recent developments in half-transitive graphs, Discrete Math. 182, 1998, 219–231. [SCI] 2. Sir´aˇn,J.:Triangleˇ group representations and constructions of regular maps, Proc. London Math. Soc. 82 (3), 2001, 1–34. [SCI]

96 R. Nedela: Group Actions on Surfaces

3. Sir´aˇn,J.:Triangleˇ group representations and their applications to graphs and maps, Discrete Math. 229 (1-3), 2001, 341–358. [SCI] 4. Sir´aˇn,J.:“Regularˇ maps”. CoM2MaC Lecture Notes (2), 2001. [M]

5. Gross,J.L.-Yellen,J.: “Handbook of Graph Theory”, CRC Press, Boca Raton, 2004. page 712. [M]

6. Li,C.H.-Sir´aˇn,J.,Regularˇ maps whose groups do not act faithfully on vertices, edges, or faces, European J. Combin. 26, 2005, 521–541. [SCI]

Nedela,R.: Regular maps - combinatorial objects relating diﬀerent ﬁelds of mathematics. Mathematics in the new millennium (Seoul, 2000). J. Korean Math. Soc. 38 (5), 2001, 1069–1105. Cited in: 2+1+1

1. Kwak,J.H.: Tesselations and regular maps, Commun. Korean Math. Soc. 18 (2003), 1–20. (Korean) 2. Lando,S.K.-Zvonkin,A.: “Graphs on Surfaces and their Applications”, Se- ries : Encyclopaedia of Mathematical Sciences , Vol. 141, Springer Verlag, Heidelberg, Gamkrelidze, R.V.; Vassiliev, V.A. (Eds.) 2004, XV. [M]

3. Kwon,Y.S.: New regular embeddings of n-cubes Qn, J. Graph Theory 46, 2004, 297–312. [SCI] 4. Li,C.H.-Sir´aˇn,J.:ˇ Regular maps whose groups do not act faithfully on vertices, edges, or faces, European J. Combin. 26, 2005, 521–541. [SCI] 5. Kwak,J.H.-Kwon,Y.S.-Feng,R.Q.: A classiﬁcation of regular t-balanced Cayley maps on dihedral groups, European J. Combin., in press. [SCI]

Nedela,R.-Skoviera,M.:ˇ Cayley snarks and almost simple groups, Com- binatorica 21, 2001, 583–590. Cited in: 1+0+1 1. Potoˇcnik,P.: Edge-colourings of cubic graphs admitting a solvable vertex- transitive group of automorphisms, Univ. Ljubljana, Preprint Ser. 39 2001, No. 781.

2. Potoˇcnik,P.: Edge-colourings of cubic graphs admitting a solvable vertex- transitive group of automorphisms, J. Combin. Theory B 91 (2), 2004, 289–300. [SCI]

Nedela,R .-Skoviera,M.-Zlatoˇs,A.:Bipartiteˇ maps, Petrie duality and exponent groups, Collection of papers published in honour of M. Pez- zana, edited by C. Galliardi, Atti Sem. Mat. Fis. Univ. Modena, Suppl. Vol 49 2001, 109–133. Cited in: 1+0+1 1. Kwak,J.H.-Kwon,Y.S.: Exponent and switch exponent groups of regular orientable embeddings of complete bipartite graphs, CoM2MaC Preprint Ser., 2003, No.4.

97 R. Nedela: Group Actions on Surfaces

2. Kwon, Y.S.: New regular embeddings of n-cubes Qn, J. Graph Theory 46 (4), 2004, 297–312. [SCI]

Marusiˇc,D.-Nedela,R.: On the point stabilizers of transitive groups with non-self-paired suborbits of length 2, J. Group Theory 4 (3), 2001, 19–43. Cited in: 3+0+0

1. Oh,J.M.-Hwang,K.W.: Construction of one-regular graphs of valency 4 and 6, Discrete Math. 278, 2004, 195–207. [SCI]

2. Wilson,S.: Semi-transitive graphs, J. Graph Theory 45 (1) 2004, 1–27. [SCI]

3. Fang,X.G.-Li,C.H.-Xu,M.Y.:On edge-transitive Cayley graphs of valency four, European J. Combin. 25, 2004, 1107–1116. [SCI]

Nedela,R.-Skoviera,M.-Zlatoˇs,A.:ˇ Regular embeddings of complete bipartite graphs, Discrete Math. 258, 2002, 379–381. Cited in: 1+0+2

1. Jones,G.: Graphs, groups and surfaces, Rendiconti del Seminario di Mat. Messina, Ser. II, Suppl. No. 8, 2002, 71–85.

2. Kwon, Y.S.: New regular embeddings of n-cubes Qn, J. Graph Theory 46 (4), 2004, 297–312. [SCI] 3. Kwak,J.H.-Kwon,Y.S.: Exponent and switch exponent groups of regular orientable embeddings of complete bipartite graphs, CoM2MaC Preprint Ser., 2003, No.4.

Malniˇc,A.-Nedela,R.-Skoviera,M.:ˇ Regular homomorphisms and Reg- ular Maps , European J. Combin. 23 (4), 2002, 449–461. Cited in: 0+0+1

1. Kwak,J.H.-Kwon,Y.S.: Generalized Cayley maps, CoM2MaC Preprint Ser., 2003, No.1.

Du,S.F.-Kwak,J.H.-Nedela,R.: Regular maps with pq vertices, J. Al- gebraic Combin. 19, 2004, 123–141. Cited in: 1+0+0

1. Kwon,Y.S.: New regular embeddings of n-cubes Qn, J. Graph Theory 46, 2004, 297–312. [SCI]

Du,S.F.-Kwak,J.H.-Nedela,R.: Regular embeddings of complete multipartite graphs, European J. Combin. 26, 2005, 505–519. Cited in: 1+0+0

1. Kwon,Y.S.: New regular embeddings of n-cubes Qn, J. Graph Theory 46, 2004, 297–312. [SCI]

98 R. Nedela: Group Actions on Surfaces

Du,S.F.-Kwak,J.H.-Nedela,R.: Regular embeddings of hypercubes of odd dimension, European J. Combin., manuscript. Cited in: 1+0+0

1. Kwon,Y.S.: New regular embeddings of n-cubes Qn, J. Graph Theory 46, 2004, 297–312. [SCI]

Jones,G.-Nedela,R.-Skoviera,M.:ˇ Regular embeddings of Kn,n where n is an odd prime power, manuscript. Cited in: 1+0+0

1. Kwon,Y.S.: New regular embeddings of n-cubes Qn, J. Graph Theory 46, 2004, 297–312. [SCI]

Archdeacon,D.-Nedela,R.-Skoviera,M.:ˇ Maximum genus, connectivity, and Nebesk´y’stheorem, Discrete Math., accepted. Cited in: 2+0+0 1. Huang,Y.Q.: The maximum genus on a 3-vertex connected graph, Graphs and Combin. 16, 2000, 165–176. [SCI] 2. Huang,Y.Q: Maximum genus and chromatic number of graphs, Discrete Math. 271, 2003, 117–127. [SCI]

Nedela,R.-Skoviera,M.:ˇ Constructions of irreducible snarks, to appear. Cited in: 1+0+0 1. Cavicchioli,A.-Meschiari,M.-Ruini,B.-Spaggiari,F.: A survey on snarks and new results: Products, Reducibility and Computer search, J. Graph The- ory 28 (2), 1998, 57–86. [SCI]

5.3 Survey of publications and citations

Totally, the above list contains 225 citations of works of the author. These include 116 citations of works of the author in journals included in the database at ISI web of science (Science Citation Index Expanded) and 27 citations in monographs.

5.4 List of invited visits and selected talks in conferences

1987 - Conference ”Graphs and other combinatorial topics” in Eger (Hungary) 1990 - ”Third Polish conference on graph theory” in Niedzica (Poland) 1990 - Conference in graph theory in Bielefeld (Germany) 1990 - Conference in graph theory in Kiel (Germany) 1991 - Conference in graph theory in Bled (Slovenia) 1992 - Conference in graph theory in Zielona Gora (Poland)

99 R. Nedela: Group Actions on Surfaces

1992 - Conference in graph theory in Braunschweig (Germany) 1992 - Lectures at Technische Hochschule Dressden (Germany) 1993 - Lectures in number theory, Ljubljana (Slovenia) 1993 - Lecture in graph theory, Maribor (Slovenia) 1993 - Lecture at Seminar, TU Leoben (Austria) 1993 - Conference Braunschweig (Germany) 1993 - Lectures at Univ. Bielefeld (Germany) 1994 - Conference Ankaran (Slovenia) 1994 - Visiting Prof. at university Ljubljana, Course in elementary number theory for undergraduate students, (one semester) 1995 - Lectures at Univ. Ljubljana for PhD.students 1995 - Univ. Politechnico di Milano a serie of lectures TU Milano and Univ. Bologna 1995 - visiting resercher at TU Milano (1 month) 1995 - TU Leoben, action Austria-Slovakia, a serie of lectures (2 weeks) 1995 - Conference in graph theory, Bled 1996 - Banach center Warschaw, invited speaker, topic: Regular maps - a graph theoretical point of view 1996 - Krakow, Invited speaker,TU Krakow topic: Half-transitive graphs and regular maps 1998 - Univ. New Brunswick (USA), invited speaker, topic: Regular maps and residual finiteness of triangle groups 1998 - Flagstaff (USA), invited speaker Workshope SIGMAC, topic: Half- transitive graphs and regular maps 1998 - Univ. Aveiro (Portugalsko), invited speaker, title: Exponents of regular maps 1999 - Bled, Slovinsko, invited speaker, title: Half-transitive graphs of valency four, 1999 - Krakow, invited speaker, workshop in graph theory, 1999 -Jindrichov Hradec, lecture at the conference : Mathematical methods in Economics, 2000 - Soul, invited speaker: Mathematics in the new milenium, a conference under the auspices of World’s Math. Soc., title: Regular maps - combinatorial objects relating different fields of mathematics, 2001 - visiting researcher, University Aveiro (6 months), 2002 - Aveiro, workshop: Regular maps with prescribed genus. 2003 Com2Mac Mini-Workshop on Hurwitz Theory and Ramifications, Po- hang, Korea 2004 - Com2Mac Mini-Workshop on 2-type-face maps, Pohang, Korea

100 R. Nedela: Group Actions on Surfaces

2004 - Chirality index and chirality group of a map, Korean Institute for Advanced Technologies, Daejon, Korea 2004 - Regular maps of prime negative Euler characteristic, Maple conference, Simon Fraser University, Vancouver. 2004 - visiting researcher, McMaster Univ. Hamilton (7 weeks) 2000-2005 - visiting researcher, POSTECH Pohang, (8 months)

5.5 Other activities

1990 Awarded by the Union of Slovak mathematicians and physicists Grants: GAV - 1992, VEGA - 1996, VEGA - 1999, KEGA- 1999, VEGA - 2002, APVT - 2002.

Membership in scientiﬁc and editorial boards: Slovak grant agency VEGA, Slovak board for Graduate Studies in Discrete mathematics, Slovak board for evaluation of Universities, Mathematics, Member of editorial boards of journals Tatra Mountains a Acta Math Univ. M. Bellii. Member of the accreditation board of the Slovak government, Member of scientiﬁc boards of several faculties: Faculty of Natural Sciences UMB (1995-2004), School of Finance UMB (1997- 2001), Faculty of Economy UMB (1995-1996), Matej Bel University (1994-1997), Mathematical Institute SAS (2005-).

Supervisor of 5 diploma thesis, supervisor of 2 graduate students, organizer of several international workshops and conferences, founder of the workshops ‘Graph embeddings and maps on surfaces’. Lectures for graduate students at several universities: Ljubljana, Milano, Aveiro, Pohang. Since year 1984 con- tinuous teaching of undergraduate students at Matej Bel University including courses in analysis, discrete mathematics and optimization.

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Chapter 6

Appendix: Reprints of papers

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