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Group Actions on Graphs, Maps and Surfaces

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Group Actions on Graphs, Maps and Surfaces Ministry of Education of Slovak Republic Scienti¯c Board of P. J. Saf¶arikUniversity· 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 Ko·sice,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, classi¯cation 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 Classi¯cation 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 ¯niteness 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 ¯xed surface . 48 3.8 Operations on maps and hypermaps, external symmetries of hy- permaps . 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 sub- orbits of length 2 (with D. Maru·si·c) . 103 Exponents of orientable maps (with M. Skoviera)· . 103 Classi¯cation of regular maps of prime negative Euler characteristic (with A. Breda and J. Sir¶a·n)· . 103 K-minimal triangulations of surfaces (with A. Malni·c) . 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 uni¯ed by the interaction of graphs, groups and surfaces. In the ¯rst 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 Maru·si·c(Univ. Ljubljana) during years 1994-2000. The strongest result we have achieved is a classi¯cation 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 ¯eld. 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 ¯eld 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 Malni·c,Alexander Mednykh, Martin Skoviera· and Jozef Sir¶a·n.· Among many interesting results I have chosen (for the Appendix) a recent paper by Breda, Nedela and Sir¶a·nclassifying· regular maps on surfaces of prime negative Euler characteristic. As a consequence, an in¯nite 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 Skoviera· 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 classi¯cation of regular embeddings of graphs. Chapter 4 deals with minimal triangulations of given edge-width. In collab- oration with Aleksander Malni·cwe 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 ¯nite basis with respect to the vertex-splitting operation. Our result generalises the well-known statement of Steinitz on polyhedral triangula- tions, 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.· 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 ¯nite 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 speci¯ed otherwise. By a directed graph X we mean an ordered pair (V; A), where V is a ¯nite 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 = f(x; y); (y; x)j[x; y] 2 E(X)g. An automorphism of a graph X is a permutation à of V such that [Ã(x);Ã(y)] 2 E(X) for every edge [x; y] 2 E(X). Similarly, an automorphism of an oriented graph X is a permutation à of V such that [Ã(x);Ã(y)] 2 A(X) for every arc [x; y] 2 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 ¯nite. For graph-theoretic and group-theoretic terms not de¯ned 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 ½ 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 O¹ denotes the complementary orbit to O then á ¡! X = (V; O¹) 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 ¡! á 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 di®erent (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 di®erent G-orientations of C4 £ C4.
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