Faster 3-colouring algorithm for graphs of diameter 3

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