Efficient Algorithms for Weighted Colorings of Series-Parallel Graphs

Abstract

Let G be a weighted graph such that each vertex v has a positive integer weight $w\left( v \right)$ . A weighted coloring of G is to assign a set of $w\left( v \right)$ colors to each vertex $v$ 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 $O\left( {nw_{max} } \right)$ , where n is the number of vertices and $w_{max}$ is the maximum vertex-weight of G.