A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

Abstract

Abstract For a positive integer k , a k ‐ coloring of a graph is a mapping such that whenever . The Coloring problem is to decide, for a given G and k , whether a k ‐coloring of G exists. If k is fixed (i.e., it is not part of the input), we have the decision problem k ‐ Coloring instead. We survey known results on the computational complexity of Coloring and k ‐ Coloring for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex.