We show that given an $n$-vertex graph $G$ of diameter 3 we can decide if $G$ is $3$-colourable in time $2^{O(n^{2/3-\varepsilon})}$ for any $\varepsilon < 1/33$. This improves on the previous best algorithm of $2^{O((n\log n)^{2/3})}$ from Dębski, Piecyk and Rzążewski [Faster 3-coloring of small-diameter graphs, ESA 2021].

翻译：我们证明，对于任意$\varepsilon < 1/33$，给定一个包含$n$个顶点的直径3图$G$，我们可以在$2^{O(n^{2/3-\varepsilon})}$时间内判定$G$是否可三着色。这改进了Dębski、Piecyk和Rzążewski在[Faster 3-coloring of small-diameter graphs, ESA 2021]中提出的$2^{O((n\log n)^{2/3})}$时间算法。