Average Cayley Genus for Cayley Maps of Dihedral Groups Generated by Their Reflections

Abstract

Let $\mathrm{\Gamma }$ be a finite group and let $\mathrm{\Delta }$ be a generating set for $\mathrm{\Gamma }$. A Cayley map associated with $\mathrm{\Gamma }$ and $\mathrm{\Delta }$ is an oriented 2-cell embedding 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 formula for the average Cayley genus is known for the dihedral group with generating set consisting of all the reflections. However, the known formula involves sums of certain coefficients of a generating function and its format does not specifically indicate the Cayley genus distribution. We determine a simplified formula for this average Cayley genus as well as provide improved understanding of the Cayley genus distribution.