Cost Total Colorings of Trees

A total coloring of a graph G is to color all vertices and edges of G so that no two adjacent or incident elements receive the same color. Let C be a set of colors, and let ω be a cost function which assigns to each color c in C a real number ω(c) as a cost of c. A total coloring f of G is called an optimal total coloring if the sum of costs ω( f (x)) of colors f (x) assigned to all vertices and edges x is as small as possible. In this paper, we give an algorithm to find an optimal total coloring of any tree T in time O(n∆3) where n is the number of vertices in T and ∆ is the maximum degree of T . key words: cost total coloring, dynamic programming, matching, total coloring, tree