New methods for finding minimum genus embeddings of graphs on orientable and non-orientable surfaces

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. The authors are very grateful to Tomaž Pisanski for suggesting that they extend their initial development of the orbit method on the Hoffman-Singleton graph to other graphs, which then led to them to develop the other methods presented here, in order to find the answers to many open questions. The authors are also grateful to the referee for some helpful suggestions about presenting their work. The first author is grateful to the N.Z. Marsden Fund for its support (grant UOA 1626), and acknowledges the use of the MAGMA system [2] for computational experiments and verification of a number of discoveries announced in this paper, as well as Sage [45] in combination with IBM CPLEX for a small number of the ILP computations. The second author acknowledges partial support from the Spanish MEC project ICWT (TIN2016-80250-R) and ARES (CONSOLIDER INGENIO 2010). E-mail addresses: m.conder@auckland.ac.nz (Marston Conder), klara.stokes@mu.ie (Klara Stokes) c b This work is licensed under https://creativecommons.org/licenses/by/4.0/

A Practical Algorithm for the Computation of the Genus

“…In [7] the genus (and also non-orientable genus) of several graphs was determined. They describe four specialized approaches they apply to some special graphs that have in general a large symmetry group.…”

“…Some of the other examples they give can also be solved and sometimes extended by our general approach without any manual interference -just by piping the graph into multi genus. In [7] in general no precise running times are given. The instances for which the results could be confirmed and sometimes extended are:…”

“…The graph C 3 C 3 C 3 : In [7] it says that with a natural vertex labeling the subgroup orbit method takes only a couple of minutes to find a genus 7 embedding. The method described here takes -depending on the labeling -from 0.19 seconds to 281 seconds to find an embedding.…”

“…The Tutte graph (or (3, 8)-cage): In [7] no running times are given, but they construct a genus 4 embedding with cyclic automorphism group of order 3. The present approach takes 0.005 seconds to determine the genus as 4 and 0.13 seconds to construct all 13, 440 embeddings.…”

“…The Gray graph: The running times for the Gray graph were already given. In [7] it is reported that there are minimum genus embeddings with an automorphism group of order 6. Checking all possible embeddings, the result is that there are 186 non-isomorphic embeddings with trivial symmetry, 23 with group size 2, 4 with group size 3, 4 with group size 6, and 2 with group size 18.…”