On the edge coloring of graph products

All graphs under consideration are nonnull, finite, undirected, and simple graphs. We adopt the standard notations dG(v) for the degree of the vertex v in the graph G, and ∆(G) for the maximum degree of the vertices of G. The edge chromatic number, χ′(G), of G is the minimum number of colors required to color the edges of G in such a way that no two adjacent edges have the same color. A graph is called a k-regular graph if the degree of each vertex is k. A cycle of a graph G is said to be Hamiltonian if it passes by all the vertices of G. A sequence F1,F2, . . . ,Fn of pairwise edge disjoint graphs with union G is called a decomposition of G and we write G=⋃ni=1Fi. In addition, if the subgraphs Fi are k-regular spanning of G, then G is called a k-factorable graph and each Fi is called a k-factor. Moreover, if Fi is Hamiltonian cycle for each i = 1,2, . . . ,n, then G is called a Hamiltonian decomposable graph. A graph M is a matching if ∆(M) = 1, and a perfect matching if the degree of each vertex is 1. An independent set of edges is a subset of E(G) in which no two edges are adjacent. Vizing [8] classified graphs into two classes, 1 and 2; a graph G is of class 1 if χ′(G) = ∆(G), and of class 2 if χ′(G) = ∆(G) + 1. It is known that a bipartite graph is of class 1. Also, a 2rregular graph is 2-factorable. It is elementary from the definitions that a graph is regular and of class 1 if and only if it is 1-factorable. Let G= (V(G),E(G)) and H = (V(H),E(H)) be two graphs. (1) The direct product G∧H has vertex set V(G∧H) = V(G)×V(H) and edge set E(G∧H) = {(u1,v1)(u2,v2) | u1u2 ∈ E(G) and v1v2 ∈ E(H)}. (2) The Cartesian product G×H has vertex set V(G×H) = V(G)×V(H) and edge set E(G×H) = {(u1,v1)(u2,v2) | u1u2 ∈ E(G) and v1 = v2, or u1 = u2 and v1v2 ∈ E(H)}.