Faster 3-coloring of small-diameter graphs

TL;DR

This paper presents a new algorithm that improves the time complexity for solving 3-coloring in small-diameter graphs, specifically diameter-2 graphs, advancing the understanding of the problem's computational limits.

Contribution

The paper introduces the first algorithm that improves upon previous subexponential solutions for 3-coloring diameter-2 graphs without additional restrictions.

Findings

Achieves 3-coloring in diameter-2 graphs in time 2^{O(n^{1/3} log^2 n)}

Provides a new combinatorial insight using probabilistic methods

Extends algorithms to list homomorphism problems for small-diameter graphs

Abstract

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.