Combinatorial coloring of 3-colorable graphs

TL;DR

This paper introduces a new combinatorial algorithm that reduces the number of colors needed to color 3-colorable graphs in polynomial time, improving upon previous bounds and combining well with semi-definite methods.

Contribution

It presents the first combinatorial improvement over Blum's 1990 bound and further reduces the coloring bound by integrating with semi-definite approaches.

Findings

Achieved a coloring bound of $ O(n^{4/11})$ colors.

Improved the best known combinatorial bound from Blum's $ O(n^{3/8})$.

Reduced the semi-definite based coloring bound to $O(n^{0.2038})$ colors.

Abstract

We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. We present a combinatorial algorithm getting down to $\tO(n^{4/11})$ colors. This is the first combinatorial improvement of Blum's $\tO(n^{3/8})$ bound from FOCS'90. Like Blum's algorithm, our new algorithm composes nicely with recent semi-definite approaches. The current best bound is $O(n^{0.2072})$ colors by Chlamtac from FOCS'07. We now bring it down to $O(n^{0.2038})$ colors.