Complexity of Ck-Coloring in Hereditary Classes of Graphs

Abstract

For a graph F, a graph G is F-free if it does not contain an induced subgraph isomorphic to F. For two graphs G and H, an H-coloring of G is a mapping f : V (G) --> V (H) such that for every edge uv E(G) it holds that f(u)f(v) E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of Pt-free graphs. We show that for every odd k ≥ 5 the Ck-Coloring problem, even in the precoloring-extension variant, can be solved in polynomial time in P9-free graphs. On the other hand, we prove that the extension version of Ck-Coloring is NP-complete for F-free graphs whenever some component of F is not a subgraph of a subdivided claw.Supported by NSF grant DMS-1763817. This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under grant number W911NF-16-1-0404. This research is supported by the National Natural Science Foundation of China (11801284). This material is based upon work supported by the National Science Foundation under Award No. DMS1802201