Abstract
The question of how to find the smallest genus of all embeddings of a given finite connected graph on an orientable (or non-orientable) surface has a long and interesting history. In this paper we introduce four new approaches to help answer this question, in both the orientable and non-orientable cases. One approach involves taking orbits of subgroups of the automorphism group on cycles of particular lengths in the graph as candidates for subsets of the faces of an embedding. Another uses properties of an auxiliary graph defined in terms of compatibility of these cycles. We also present two methods that make use of integer linear programming, to help determine bounds for the minimum genus, and to find minimum genus embeddings. This work was motivated by the problem of finding the minimum genus of the Hoffman-Singleton graph, and succeeded not only in solving that problem but also in answering several other open questions.
Highlights
The question of how to find the smallest genus of those embeddings of a given finite connected graph on an orientable surface is a natural extension of determining whether or not a graph is planar, and has a long and interesting history
Youngs gave the first proof of the () well known fact that every orientable embedding of a connected graph is determined by the rotations of edges at its vertices [52], and this was taken further by Duke [12] to show that the range of genera of embeddings of a given connected finite graph is an unbroken sequence of non-negative integers
Our method finds minimum genus embeddings for which some non-trivial subgroup of the automorphism group of the graph induces a group of automorphisms of the map, usually with a small number of orbits on faces, when such a subgroup exists
Summary
The question of how to find the smallest genus of those embeddings of a given finite connected graph on an orientable (or non-orientable) surface is a natural extension of determining whether or not a graph is planar, and has a long and interesting history. The Hoffman-Singleton graph is the unique Moore graph of valency 7 and diameter 2 (and the largest known Moore graph of diameter 2), and is a 7-valent connected graph of order 50, diameter 2 and girth 5 The properties of this graph, including its order and valency, made it challenging to find the minimum genus using existing methods (as summarised in [50] for example), and so we had to take a new approach. Our third approach uses (mixed) integer linear programming to achieve the same thing when the auxiliary graph method is not helpful, and our fourth method uses integer linear programming directly for finding the faces of a minimum genus embedding of the graph All of these methods are quite general, in the sense that they do not expect the given graph to possess some non-trivial symmetry, even though we developed each of them to deal with graphs that do.
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