42 research outputs found
Derived voltage graphs come from an adjunction
We prove that the notion of a derived voltage graph comes from an adjunction
between the category of voltage graphs and a category of group labeled graphs
New methods for finding minimum genus embeddings of graphs on orientable and non-orientable surfaces
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