Complexity of the packing coloring problem for trees

Packing coloring is a partitioning of the vertex set of a graph with the property that vertices in the i-th class have pairwise distance greater than i. The main result of this paper is a solution of an open problem of Goddard et al. showing that the decision whether a tree allows a packing coloring with at most k classes is NP-complete. We further discuss specific cases when this problem allows an efficient algorithm. Namely, we show that it is decideable in polynomial time for graphs of bounded treewidth and diameter, and fixed parameter tractable for chordal graphs. We accompany these results by several observations on a closely related variant of the packing coloring problem, where the lower bounds on the distances between vertices inside color classes are determined by an infinite nondecreasing sequence of bounded integers. (c) 2008 Elsevier B.V. All rights reserved.