The hypocoloring problem: Complexity and approximability results when the chromatic number is small

We consider a weighted version of the subcoloring problem that we call the hypocoloring problem: given a weighted graph G (V, E; w) where w (nu) greater than or equal to 0, the goal consists in finding a partition S (S-1,...,S-k) of the node set of G into hypostable sets and minimizing Sigma(i-1)(k) w(S-i) where an hypostable S is a subset of nodes which generates a collection of node disjoint cliques K. The weight of S is defined as max{Sigma(nuis an element ofK) w(nu)\ K is an element of S}. Properties of hypocolorings are stated; complexity and approximability results are presented in some graph classes. The associated decision problem is shown to be NP-complete for bipartite graphs and triangle-free planar graphs with maximum degree 3. Polynomial algorithms are given for graphs with maximum degree 2 and for trees with maximum degree Delta.