Highly symmetric embeddings of graphs on surfaces

Reade, Olivia (2023). Highly symmetric embeddings of graphs on surfaces. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.000159d4

Abstract

This thesis considers highly symmetric maps, that is embeddings of graphs in surfaces such that the automorphism group is “large”. This may be when the automorphism group of the map acts regularly on the flag-set of the map, as for the fully regular maps studied in Part I. In contrast, Part II focusses on a class of maps where the automorphism group has (up to) two orbits on the flag-set and may not be edge-transitive.

Part I is dedicated to advancing the understanding of fully regular maps with external symmetries. Chapter 2 proves that for arbitrary valency greater than three, a fully regular map with Trinity symmetry exists, extending the previously-known existence of such a map for every even valency. Chapter 3 addresses a group of operators which acts on fully regular maps whose automorphism group is isomorphic to SL(2, 2^α). The group of operators, which depends on the value of α and is defined more precisely in Chapter 3, includes the dual and Petrie operators as well as the allowable hole operators. One approach is by exploring the orbits of this group as it acts on the space of all maps with automorphism group isomorphic to SL(2, 2^α) for the given α. A detailed investigation is presented for the group of operators acting on the set consisting of all maps with automorphism group A5 which is isomorphic to SL(2, 4).

In Part II, the focus is on edge-biregular maps. These maps can be identified with group presentations which have a particular form, namely they are generated by four involutions which partition into two distinct sets each consisting of a pair of commuting involutions. Edge-biregular maps correspond to the most symmetric examples of maps with bipartite medial graph. By the definition, each edge-biregular map inherits a two-colouring on the edges, and so long as the map is not degenerate in some way, both the valency and the face length are even. In Chapter 4 these maps are introduced, foundations are laid and degeneracies are addressed. Chapter 5 is a partial classification covering edge-biregular maps whose colour-preserving automorphism group is dihedral, and/or whose surface has Euler characteristic which is either non-negative or negative and prime. The context for Chapter 6 is edge-biregular maps whose underlying group is symmetric or alternating. A genuinely edge-biregular map is an edge-biregular map which (when disregarding the colouring of edges) is not a fully regular map. The chapter includes a proof that, with the exception of some small cases, a genuinely edge-biregular map of every feasible type exists such that the colour preserving automorphism group is symmetric or alternating.

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