New results on generalized graph coloring

Abstract

For graph classes P1, ..., Pk, Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph G can be partitioned into subsets V1, ..., Vk so that V j induces a graph in the class Pj (j = 1, 2, ..., k). If P1 = ⋯ = Pk is the class of edgeless graphs, then this problem coincides with the standard vertex k-COLORABILITY, which is known to be NP-complete for any k ≥ 3. Recently, this result has been generalized by showing that if all Pi's are additive hereditary, then the generalized graph coloring is NP-hard, with the only exception of bipartite graphs. Clearly, a similar result follows when all the Pi's are co-additive. In this paper, we study the problem where we have a mixture of additive and co-additive classes, presenting several new results dealing both with NP-hard and polynomial-time solvable instances of the problem. © 2004 Discrete Mathematics and Theoretical Computer Science (DMTCS).