Finite commutative rings whose unitary Cayley graphs have positive genus

Abstract

The unitary Cayley graph of a ring $R$ is the simple graph whose vertices are the elements of $R$, and where two distinct vertices $x$ and $y$ are linked by an edge if and only if $x-y$ is a unit in $R$. The genus of a simple graph $G$ is the smallest nonnegative integer $g$ such that $G$ can be embedded into an orientable surface $\mathbb {S}_{g}$. It is proven that, for a given positive integer $g$, there are at most finitely many finite commutative rings whose unitary Cayley graphs have genus $g$. We determine all finite commutative rings whose unitary Cayley graphs have genus 1, 2 and 3, respectively.