The optimum cost chromatic partition problem

In this paper, we study the optimum cost chromatic partition (OCCP) problem for several graph classes. The OCCP problem is the problem of coloring the vertices of a graph such that adjacent vertices get different colors and that the total coloring costs are minimum. First, we prove that the OCCP problem graphs with constant treewidth k can be solved in O(\V\ . (log\V\)(k+1)) time, respectively. Next, we study an ILP formulation of the OCCP problem given by Sen et al. [9]. We show that the corresponding polyhedron contains only integral 0/1 extrema if and only if the graph G is a diamond - free chordal graph. Furthermore, we prove that the OCCP problem is NP-complete for bipartite graphs. Finally, we show that the precoloring extension and the OCCP problem are NP-complete for permutation graphs.