Faster 3-colouring algorithm for graphs of diameter 3

TL;DR

This paper presents a significantly faster algorithm for determining 3-colorability of graphs with diameter 3, improving the computational complexity over previous methods for large graphs.

Contribution

The authors develop a new algorithm that decides 3-colorability of diameter-3 graphs in subexponential time, surpassing prior algorithms' efficiency.

Findings

Decides 3-colorability in time 2^{O(n^{2/3-\varepsilon})} for any \\varepsilon < 1/33

Improves upon the previous best algorithm with time 2^{O((n \\log n)^{2/3})}

Advances understanding of graph coloring complexity for small-diameter graphs.

Abstract

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 D\k{e}bski, Piecyk and Rz\k{a}\.zewski [Faster 3-coloring of small-diameter graphs, ESA 2021].