### New Results on Generalized Graph Coloring

*Vladimir E. Alekseev, Alastair Farrugia, Vadim V. Lozin*

#### Abstract

For graph classes ℘1,...,℘k, 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 ℘j (j=1,2,...,k). If ℘1=...=℘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 ℘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 ℘i's are co-additive.

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