The Chromatic Index of an Abelian Cayley Graph

Suppose \(\Gamma\) is a finite multiplicative group and \(S \subseteq \Gamma\) satisfies \(1 \notin S\) and \(S^{-1} = \{x^{-1} | x \in S\} = S\). The abelian Cayley graph \(G = G(\Gamma, S)\) is the simple graph having vertex set \(V(G) = \Gamma\), 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 \(G = G(\Gamma, S)\): if \(\langle S \rangle\) denotes the subgroup generated by \(S\), then \(\chi'(G) = \Delta(G)\) if and only if \(|\langle S \rangle|\) is even.