Faster 3-coloring of small-diameter graphs

We study the 3-\textsc{Coloring} problem in graphs with small diameter. In 2013, Mertzios and Spirakis showed that for $n$-vertex diameter-2 graphs this problem can be solved in subexponential time $2^{\mathcal{O}(\sqrt{n \log n})}$. Whether the problem can be solved in polynomial time remains a well-known open question in the area of algorithmic graphs theory. In this paper we present an algorithm that solves 3-\textsc{Coloring} in $n$-vertex diameter-2 graphs in time $2^{\mathcal{O}(n^{1/3} \log^{2} n)}$. This is the first improvement upon the algorithm of Mertzios and Spirakis in the general case, i.e., without putting any further restrictions on the instance graph. In addition to standard branchings and reducing the problem to an instance of 2-\textsc{Sat}, the crucial building block of our algorithm is a combinatorial observation about 3-colorable diameter-2 graphs, which is proven using a probabilistic argument. As a side result, we show that 3-\textsc{Coloring} can be solved in time $2^{\mathcal{O}( (n \log n)^{2/3})}$ in $n$-vertex diameter-3 graphs. We also generalize our algorithms to the problem of finding a list homomorphism from a small-diameter graph to a cycle.