Faster 3-Coloring of Small-Diameter Graphs

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