  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.
