On the Genus of Finite Abelian Groups

Abstract

The genus of a group is the minimum genus for any Cayley color graph of the group. Using the structure theorem for finite abelian groups and appropriate current graphs, we construct quadrilateral embeddings of minimum degree for such groups in almost all cases where this is consistent with the Euler formula, thereby determining the genus for these groups, in the majority of the cases considered. In the process, graph genus embeddings are obtained for certain repeated cartesian products of cycles.