For a fixed graph $H$, in the graph homomorphism problem, denoted by $Hom(H)$, we are given a graph $G$ and we have to determine whether there exists an edge-preserving mapping $\varphi: V(G) \to V(H)$. Note that $Hom(C_3)$, where $C_3$ is the cycle of length $3$, is equivalent to $3$-Coloring. The question whether $3$-Coloring is polynomial-time solvable on diameter-$2$ graphs is a well-known open problem. In this paper we study the $Hom(C_{2k+1})$ problem on bounded-diameter graphs for $k\geq 2$, so we consider all other odd cycles than $C_3$. We prove that for $k\geq 2$, the $Hom(C_{2k+1})$ problem is polynomial-time solvable on diameter-$(k+1)$ graphs -- note that such a result for $k=1$ would be precisely a polynomial-time algorithm for $3$-Coloring of diameter-$2$ graphs. Furthermore, we give subexponential-time algorithms for diameter-$(k+2)$ and -$(k+3)$ graphs. We complement these results with a lower bound for diameter-$(2k+2)$ graphs -- in this class of graphs the $Hom(C_{2k+1})$ problem is NP-hard and cannot be solved in subexponential-time, unless the ETH fails. Finally, we consider another direction of generalizing $3$-Coloring on diameter-$2$ graphs. We consider other target graphs $H$ than odd cycles but we restrict ourselves to diameter $2$. We show that if $H$ is triangle-free, then $Hom(H)$ is polynomial-time solvable on diameter-$2$ graphs.

翻译：对于固定图$H$，在图同态问题$Hom(H)$中，我们给定图$G$，需判断是否存在边保持映射$\varphi: V(G) \to V(H)$。注意到$Hom(C_3)$（其中$C_3$为长度为3的圈）等价于3-着色问题。在直径2的图上能否多项式时间求解3-着色是一个著名的开放问题。本文研究$k\geq 2$时，有界直径图上的$Hom(C_{2k+1})$问题（即考虑除$C_3$外的所有奇圈）。我们证明对于$k\geq 2$，$Hom(C_{2k+1})$问题在直径$(k+1)$图上可多项式时间求解——需注意，当$k=1$时该结论将直接给出直径2图上3-着色的多项式时间算法。此外，我们针对直径$(k+2)$和$(k+3)$图给出亚指数时间算法。我们通过直径$(2k+2)$图的下界来补充上述结果：在此类图中$Hom(C_{2k+1})$问题为NP难问题，且除非ETH失效否则无法在亚指数时间内求解。最后，我们考虑直径2图上3-着色的另一推广方向：将目标图$H$从奇圈扩展至其他图，但限制在直径2的图类中。我们证明若$H$不含三角形，则$Hom(H)$在直径2图上可多项式时间求解。