Efficient algorithms for weighted colorings of series-parallel graphs

Let G be a weighted graph such that each vertex v has a positive integer weight omega(nu). A weighted coloring of G is to assign a set of omega(nu) colors to each vertex nu so that any two adjacent vertices receive disjoint sets of colors. This paper gives an efficient algorithm to find the minimum number of colors required for a weighted coloring of a given series-parallel graph G in time O(nomega(max)), where n is the number of vertices and omega(max) is the maximum vertex-weight of G.