Toroidal involutory Cayley graphs

Abstract

Suppose that $R$ is a finite commutative ring with identity. The involutory Cayley graph $\G(R)$ of $R$ is the graph whose vertices are the elements of $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $(x-y{)}^2=1$. In this paper, we classify all rings $R$ for which $\G(R)$ is a toroidal graph, that is, a graph that can be embedded on a torus.