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，给定一个图$G$，我们需要判断是否存在一个保边映射$\varphi: V(G) \to V(H)$。注意$Hom(C_3)$（其中$C_3$为长度为3的环）等价于3-染色问题。直径2图上的3-染色问题是否能在多项式时间内求解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$C_3$之外的所有奇环。我们证明当$k\geq 2$时，$Hom(C_{2k+1})$问题在直径$(k+1)$图上可在多项式时间内求解——值得注意的是，对于$k=1$，这样的结果恰恰是直径2图上3-染色问题的多项式时间算法。此外，我们给出了直径$(k+2)$和$(k+3)$图的亚指数时间算法。我们通过直径$(2k+2)$图的下界来补充这些结果——在此类图类中，$Hom(C_{2k+1})$问题是NP难的，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-染色问题的另一个推广方向。我们考虑除奇环外的其他目标图$H$，但将范围限制在直径2图上。我们证明若$H$不含三角形，则$Hom(H)$在直径2图上可在多项式时间内求解。