  For a graph $F$, a graph $G$ is \emph{$F$-free} if it does not contain an
induced subgraph isomorphic to $F$. For two graphs $G$ and $H$, an
\emph{$H$-coloring} of $G$ is a mapping $f:V(G)\rightarrow V(H)$ such that for
every edge $uv\in E(G)$ it holds that $f(u)f(v)\in E(H)$. We are interested in
the complexity of the problem $H$-{\sc Coloring}, which asks for the existence
of an $H$-coloring of an input graph $G$. In particular, we consider $H$-{\sc
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-{\sc Coloring} of $P_t$-free
graphs.
  We show that for every odd $k \geq 5$ the $C_k$-{\sc 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 $C_k$-{\sc 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 $C_k$-{\sc Coloring} for every odd $k \geq 5$,
b) list version of $C_k$-{\sc Coloring} for every even $k \geq 6$.
