New Results on Generalized Graph Coloring

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 Vj 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 induced-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 NPhard and polynomial-time solvable instances of the problem.