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 $k \geq 6$.

翻译：对于图$F$，若图$G$不包含与$F$同构的诱导子图，则称$G$为\emph{$F$-自由图}。对于两个图$G$和$H$，$G$的一个\emph{$H$-着色}是映射$f:V(G)\rightarrow V(H)$，使得对每条边$uv\in E(G)$，均有$f(u)f(v)\in E(H)$。我们关注问题$H$-{\sc Coloring}的复杂性，该问题询问输入图$G$是否存在$H$-着色。特别地，我们考虑$F$-自由图上的$H$-{\sc Coloring}，其中$F$是固定图，$H$是长度至少为5的奇环。该问题与著名的开放问题——确定$P_t$-自由图上3-{\sc Coloring}的复杂性密切相关。我们证明，对于每个奇数$k \geq 5$，$C_k$-{\sc Coloring}问题（甚至其列表变体）可在$P_9$-自由图上多项式时间内求解。该算法可推广至$k$为长度至少10的偶数时的$C_k$-{\sc Coloring}列表版本。另一方面，我们证明若$F$的某个连通分支不是细分爪图的子图，则以下问题在$F$-自由图上是NP完全的：a) 每个奇数$k \geq 5$的$C_k$-{\sc Coloring}扩展版本，b) 每个偶数$k \geq 6$的$C_k$-{\sc Coloring}列表版本。