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 Z(k) so that the induced vertex k-coloring, in which the color of a vertex v in G is the sum in 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 >= 4, and the square C-n(2), n >= 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 square P-n is determined, where square denotes the Cartesian product. C-r and P-r are, respectively, the cycle, and the path on r vertices each.