Twin edge colorings of certain square graphs and product graphs

A twin edge k-coloring of a graph G is a proper edge k-coloring of G with the elements of Zk sothat the induced vertex k-coloring, in which the color of a vertex v in G is the sum in Zk of thecolors of the edges incident with v; is a proper vertex k-coloring. The minimum k for which Ghas a twin edge k-coloring is called the twin chromatic index of G: Twin chromatic index of thesquare P2 n; n ≥ 4; and the square C2n ; n ≥ 6; are determined. In fact, the twin chromatic indexof the square C2 7 is ∆ + 2; where∆ is the maximum degree. Twin chromatic index of Cm □ Pn is determined, where 2 denotes the Cartesian product. Cr and Pr are, respectively, the cycle, and thepath on r vertices each.