Complexity of C_k-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∈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 P₉-free graphs. The algorithm extends to the list version of C_k-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: 1. the precoloring extension version of C_k-COLORING for every odd k≥5; 2. the list version of C_k-COLORING for every even k≥6.