A coloring problem for weighted graphs

Let G be a simple undirected graph and let w be an assignment of non-negative weights to the vertices of G. For a proper r-coloring c of the vertices of G, we denote by w(c)(i) the maximum weight of a vertex in color class i, and denote by w(c) the sum, Sigma(i=1)(r) w(c)(i), of these maximum weights. Let sigma(G, w, r) be the minimum of w(c) among all r-colorings of G, and let sigma(G, w) be the minimum among sigma(G, w, r) for all r greater than or equal to 1. A coloring c of a weighted graph G is called optimal coloring if w(c) = sigma(G, w). We shall consider the problem that how many colors are needed in an optimal coloring of a weighted graph Gi We relate this number to the First-Fit coloring algorithm and show that for optimal colorings of trees, the number of colors needed could be arbitrarily large, and for graphs of path-width k, the number of colors needed in an optimal coloring is at most 40(k + 1). We then give a polynomial time algorithm to decide, for a fixed r, whether or not a weighted graph G of bounded tree-width satisfies sigma(G, w, r) less than or equal to q, for any given number q. (C) 1997 Elsevier Science B.V.