New Results on Generalized Graph Coloring

For graph classes \wp_1,...,\wp_k, Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph G can be partitioned into subsets V_1,...,V_k so that V_j induces a graph in the class \wp_j (j=1,2,...,k). If \wp_1=...=\wp_k 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 \wp_i'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 \wp_i's are co-additive.