The Chromatic Index of an Abelian Cayley Graph

Abstract

Suppose $\mathrm{\Gamma }$ is a finite multiplicative group and $S\subseteq \mathrm{\Gamma }$ satisfies $1\notin S$ and $S^{-1}=\{x^{-1}|x\in S\}=S$. The abelian Cayley graph $G=G(\mathrm{\Gamma },S)$ is the simple graph having vertex set $V(G)=\mathrm{\Gamma }$, an abelian group, and edge set $E(G)=\{\{x,y\}|x^{-1}y\in S\}$. We prove the following regarding the chromatic index of an abelian Cayley graph $G=G(\mathrm{\Gamma },S)$: if $⟨S⟩$ denotes the subgroup generated by $S$, then ${\chi }^{\prime }(G)=\mathrm{\Delta }(G)$ if and only if $|⟨S⟩|$ is even.