Equitable Δ-coloring of graphs

Consider a graph G consisting of a vertex set V (G) and an edge set E(G). Let Delta(G) and chi(G) denote the maximum degree and the chromatic number of G. respectively. We say that G is equitably Delta(G)-colorable if there exists a proper Delta(G)-coloring of G such that the sizes of any two color classes differ by at most one. Obviously, if G is equitably Delta(G)-colorable, then Delta(G) >= chi (G). Conversely, even if G satisfies A (G) >= chi(G), we cannot guarantee that G must be equitably A (G)-colorable. In 1994, the Equitable Delta-Coloring Conjecture (E Delta CC) asserts that a connected graph G with Delta(G) >= chi(G) is equitably Delta(G)-colorable if G is different from K-2n+1,K-2n+1 for all n >= 1. In this paper, we give necessary conditions for a graph G (not necessarily connected) with Delta(G) >= chi (G) to be equitably Delta(G)-colorable and prove that those necessary conditions are also sufficient conditions when G is a bipartite iv (3G) I graph, or G satisfies Delta(G) >= vertical bar V(G)vertical bar/3 + 1, or G satisfies Delta(G) <= 3. (C) 2011 Elsevier B.V. All rights reserved.