For $k\geq 1$, a $k$-colouring $c$ of $G$ is a mapping from $V(G)$ to $\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ for any two non-adjacent vertices $u$ and $v$. The $k$-Colouring problem is to decide if a graph $G$ has a $k$-colouring. For a family of graphs ${\cal H}$, a graph $G$ is ${\cal H}$-free if $G$ does not contain any graph from ${\cal H}$ as an induced subgraph. Let $C_s$ be the $s$-vertex cycle. In previous work (MFCS 2019) we examined the effect of bounding the diameter on the complexity of $3$-Colouring for $(C_3,\ldots,C_s)$-free graphs and $H$-free graphs where $H$ is some polyad. Here, we prove for certain small values of $s$ that $3$-Colouring is polynomial-time solvable for $C_s$-free graphs of diameter $2$ and $(C_4,C_s)$-free graphs of diameter $2$. In fact, our results hold for the more general problem List $3$-Colouring. We complement these results with some hardness result for diameter $4$.

中文翻译：

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缺少小循环时有界直径的着色图

对于$ k \ geq 1 $，$ G $的$ k $着色$ c $是从$ V（G）$到$ \ {1,2，\ ldots，k \} $的映射，使得$ c （u）\ neq c（v）$表示任意两个非相邻顶点$ u $和$ v $。$ k $上色问题是确定图形$ G $是否具有$ k $上色。对于一系列图$ {\ cal H} $，如果$ G $不包含来自$ {\ cal H} $的任何图作为诱导子图，则图$ G $无$ {\ cal H} $。假设$ C_s $为$ s $-顶点循环。在以前的工作（MFCS 2019）中，我们研究了限制直径对$ 3 $复杂度的影响-对于$（C_3，\ ldots，C_s）$无图和$ H $无图（其中$ H $是一些Polyad。在这里，我们证明对于$ s $的某些较小值，对于直径为$ 2 $的无$ C_s $的图和直径为$ 2 $的无$（C_4，C_s）$的图，$ 3 $是可多项式时间解的。实际上，我们的结果适用于更普遍的问题List $ 3 $ -Colouring。