The maximum genus of vertex-transitive graphs

Graphs possessing a high degree of symmetry have often been considered in topological graph theory. For instance, a number of constructions of genus embeddings by means of current or voltage graphs is based on the observation that a graph can be represented as a Cayley graph for some group. Another kind of embedding problems where symmetrical graphs are encountered is connected with regular maps or symmetrical embeddings, generally in the interior of the embedding range. In this paper we focus on the other extreme of the embedding range by investigating the maximum genus of vertex-transitive graphs. A similar problem has been studied by Khomenko and Glukhov [4, Theorem lo] who showed that every edge-transitive graph is upper-embeddable. In the present paper we shall see that, by contrast, vertex-transitive graphs may not be upper-embeddable in general. However, we shall succeed in characterizing all those vertex-transitive graphs that are not upper-embeddable and in determining their maximum genus. We show that a simple connected vertex-transitive graph of valency k and girth g is upper-embeddable whenever k 2 4 or g 2 4. The remaining vertex-transitive graphs, apart from cycles and three small-order exceptions, are not upperembeddable. These coincide, however, with truncated cubic graphs (see Section 1 for the definition) whose maximum genus was computed in [l] and [9]. A particular attention is paid to Cayley graphs. It is interesting that a cubic Cayley graph K is not upper-embeddable if and only if K is the truncation of the graph underlying a 3-valent regular map on an orientable surface, with some exceptions.