Better coloring of 3-colorable graphs

TL;DR

This paper introduces a new combinatorial algorithm and combines it with SDP techniques to improve the bounds on coloring 3-colorable graphs, reducing the number of colors needed to approximately n^{0.19747}.

Contribution

It presents a novel combinatorial approach for graphs with high minimum degree and integrates it with SDP methods to improve coloring bounds for 3-colorable graphs.

Findings

Achieves coloring with approximately n^{0.19747} colors.

Improves previous bounds from n^{0.19996} to n^{0.19747}.

Provides a polynomial-time algorithm for graphs with minimum degree above √n.

Abstract

We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. This is one of the most challenging problems in graph algorithms. In this paper using Blum's notion of ``progress'', we develop a new combinatorial algorithm for the following: Given any 3-colorable graph with minimum degree $\ds>\sqrt n$, we can, in polynomial time, make progress towards a $k$-coloring for some $k=\sqrt{n/\ds}\cdot n^{o(1)}$. We balance our main result with the best-known semi-definite(SDP) approach which we use for degrees below $n^{0.605073}$. As a result, we show that $\tO(n^{0.19747})$ colors suffice for coloring 3-colorable graphs. This improves on the previous best bound of $\tO(n^{0.19996})$ by Kawarabayashi and Thorup in 2017.