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 so that the induced vertex k-coloring, in which the color of a vertex v in G is the sum in Zk 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 2 n , n ≥ 4, and the square C n, n ≥ 6, are determined. In fact, the twin chromatic index of the square C 7 is ∆ + 2, where ∆ is the maximum degree. Twin chromatic index of Cm 2Pn is determined, where 2 denotes the Cartesian product. Cr and Pr are, respectively, the cycle, and the path on r vertices each.