Complexity of $C_k$-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 P t -free graphs.We show that for every odd k ≥ 5 the C k -Coloring problem, even in the list variant, can be solved in polynomial time in P 9 -free graphs. The algorithm extends for the case of list version of C k -Coloring, where k is an even number of length at least 10.On the other hand, we prove that if some component of F is not a subgraph of a subdividecd claw, then the following problems are NP-complete in F -free graphs: a) extension version of C k -Coloring for every odd k ≥ 5, b) list version of C k -Coloring for every even k ≥ 6.