Precoloring extension. I. Interval graphs

of the following general problem on vertex colorings of graphs. Suppose that some vertices of a graph G are assigned to some colors. Can this ‘precoloring’ be extended to a proper coloring of G with at most k colors (for some given k)? This question was motivated by practical problems in scheduling and VLSI theory. Here we investigate its complexity status for interval graphs and for graphs with a bounded treewidth. 1. Introduction. We consider …nite undirected graphs G = (V;E) with vertex set V and edge set E. The clique number or maximum clique size and the chromatic number of G are denoted by !(G) and (G), respectively. For any vertex subset W V , GW denotes the subgraph induced by W . By de…nition, for a given integer k 2, a (proper) k-coloring is a function f : V ! f1; 2; : : : ; kg such that uv 2 E implies f(u) 6= f(v). The problem we raise and investigate in this paper is called the PRECOLORING EXTENSION problem, or PrExt in short. PrExt is more general than the usual CHROMATIC NUMBER problem and less general than LIST-COLORING [21]. (The latter has been studied extensively for line graphs [5,9,13,15].) PrExt can be formulated as follows: Instance. An integer k 2, a graph G = (V;E) with jV j k, a vertex subset W V , and a proper k-coloring of GW . Question. Can this k-coloring be extended to a proper k-coloring of the whole graph G?