Twin edge colorings of certain square graphs and product graphs

Abstract

A twin edge $k\!$-coloring of a graph $G$ is a proper edge $k$-coloring of $G$ with the elements of $\mathbb{Z}_k$ so that the induced vertex $k$-coloring, in which the color of a vertex $v$ in $G$ is the sum in $\mathbb{Z}_k$ of the colors of the edges incident with $v,$ is a proper vertex $k\!$-coloring. The minimum $k$ for which $G$ has a twin edge $k\!$-coloring is called the twin chromatic index of $G.$ Twin chromatic index of the square $P_n^2,$ $n\ge 4,$ and the square $C_n^2,$ $n\ge 6,$ are determined. In fact, the twin chromatic index of the square $C_7^2$ is $\Delta+2,$ where $\Delta$ is the maximum degree. Twin chromatic index of $C_m\,\Box\,P_n$ is determined, where $\Box$ denotes the Cartesian product. $C_r$ and $P_r$ are, respectively, the cycle, and the path on $r$ vertices each.