Toroidal involutory Cayley graphs

TL;DR

This paper classifies all finite commutative rings with identity whose involutory Cayley graphs can be embedded on a torus, linking algebraic structures to topological graph properties.

Contribution

It provides a complete classification of rings R where the involutory Cayley graph G(R) is toroidally embeddable, connecting ring theory with topological graph theory.

Findings

Identifies all rings R with toroidal G(R)

Characterizes the structure of R for toroidal embeddings

Links algebraic properties to topological graph embeddings

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.