# 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 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.

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