Complexity of Ck-coloring in hereditary classes of graphs

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 is an element of E(G) it holds that f (u) f (v) is an element of 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 list variant, can be solved in polynomial time in P9-free graphs. The algorithm extends to the list version of Ck-CoLoRING, where k >= 10 is an even number.On the other hand, we prove that if some component of F is not a subgraph of a subdivided claw, then the following problems are NP-complete in F-free graphs:a) the precoloring extension version of Ck-CoLoRING for every odd k >= 5;b) the list version of Ck-CoLoRING for every even k >= 6.(c) 2023 Elsevier Inc. All rights reserved.