Algorithm for the cost edge-coloring of trees

Let C be a set of colors, and let ω be a cost function which assigns a real number ω(c) to each color c in C. An edge-coloring of a graph G is to color all the edges of C so that any two adjacent edges are colored with different colors. In this paper we give an efficient algorithm to find an optimal edge-coloring of a given tree T, that is, an edge-coloring f of T such that the sum of costs ω(f(e)) of colors f(e) assigned to all edges e is minimum among all edge-colorings of T. The algorithm takes time O(nΔ²) if n is the number of vertices and Δ is the maximum degree of T.