Equitable colorings of Cartesian products of graphs

The present paper studies the following variation of vertex coloring on graphs. A graph G is equitably k-colorable if there is a mapping f : V (G) -> {1,2, . . . . k} such that f(x) not equal f(y) for xy is an element of E(G) and parallel to f(-1)(i) vertical bar - vertical bar f(-1)(j) parallel to <= 1 for 1 <= i,j <= k The equitable chromatic number of a graph G, denoted by X=(G), is the minimum k such that G is equitably k-colorable. The equitable chromatic threshold of a graph G, denoted by x*=(G), is the minimum t such that G is equitably k-colorable for all k >= t. Our focus is on the equitable colorability of Cartesian products of graphs. In particular, we give exact values or upper bounds of X=(G square H) and X*=(G square H) when G and H are cycles, paths, stars, or complete bipartite graphs. (C) 2011 Elsevier B.V. All rights reserved.