New results on generalized graph coloring

For graph classes $P_1,...,P_k$, 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 $P_j$ $(j=1,2,...,k)$. If $P_1 = ... = P_k$ is the class of edgeless graphs, then this problem coincides with the standard vertex $k$-{\sc colorability}, which is known to be NP-complete for any $k\ge 3$. Recently, this result has been generalized by showing that if all $P_i$'s are additive induced-hereditary, then generalized graph coloring is NP-hard, with the only exception of recognising bipartite graphs. Clearly, a similar result follows when all the $P_i$'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.