Twin edge colorings of certain square graphs and product graphs

R Rajarajachozhan (Department of Mathematics Annamalai University Annamalainagar 608 002, India)
R. Sampathkumar (Mathematics Section, Faculty of Engineering and Technology, Annamalai University Annamalainagar 608 002, India)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 4, No 1 (2016): Electronic Journal of Graph Theory and Applications

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.