Faster 3-Coloring of Small-Diameter Graphs

Abstract

We study the 3-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 graph theory. In this paper we present an algorithm that solves 3-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-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-Coloring can be solved in time $2^{\mathcal{O}( (n \log n)^{2/3})}$ in $n$-vertex diameter-3 graphs. This is the first algorithm for 3-Coloring which works in subexponential time for all diameter-3 graphs. We also discuss generalizations of our results to the weighted variant of 3-Coloring.