On the Likelihood of Symmetrical Cayley Maps for Certain Abelian Groups

Abstract

Let $\mathrm{\Gamma }$ be a finite group and let $\mathrm{\Delta }$ be a generating set for $\mathrm{\Gamma }$. A Cayley map is an orientable 2-cell imbedding of the Cayley graph $G_{\mathrm{\Delta }}(\mathrm{\Gamma })$ such that the rotation of arcs emanating from each vertex is determined by a unique cyclic permutation of generators and their inverses. A probability model for the set of all Cayley maps for a fixed group and generating set, where the distribution is uniform. We focus on certain finite abelian groups with generating set chosen as the standard basis. A lower bound is provided for the probability that a Cayley map for such a group and generating set is symmetrical.