From the first theorem in this paper, it is clear that a cayley graph on abelian group for all generating sets of even order is class $1$, that is can be edge colored in exactly $\Delta$ colors. But, this procedure and proof is not clear for me in the case of circulant graphs.
In the proof of the theorem $2.3$ of the above paper, the author proves the theorem for abelian groups by first proving the theorem for any graph defined on any generating set on the groups $H\rtimes G$. But, how can a graph defined on a cyclic group be written in this form? Specifically, say we have a cyclic group of order $2p$, where $p$ is $\it{odd}$. Then, though the group is isomorphic to $\mathbb{Z}_p\times\mathbb{Z}_2$, i cannot see how a graph defined on the whole group $\mathbb{Z}_{2p}$ is isomorphic to a cartesian product of graphs of the form $\Gamma(S:\mathbb{Z}_p)\times\mathbb{Z}_2$, as expected from the paper. Am I missing something? Thanks beforehand.
