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

Abstract

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(v) ≥ 0, the goal consists in finding a partition S = (S1,..., Sk) of the node set of G into hypostable sets and minimizing ∑ i=1k w(Si) 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{∑vεk w(v)| K ε 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 Δ. © Springer-Verlag 2004.