Coloring 3-Colorable Graphs with Less than n 1/5 Colors

Abstract

We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. We first present a new combinatorial algorithm using Õ ( n 4/11 ) colors. This is the first combinatorial improvement since Blum’s Õ ( n 3/8 ) bound from FOCS’90. Like Blum’s algorithm, our new algorithm composes immediately with recent semi-definite programming approaches, and improves the best bound for the polynomial time algorithm for the coloring of 3-colorable graphs from O ( n 0.2072 ) colors by Chlamtac from FOCS’07 to O ( n 0.2049 ) colors. Next, we develop a new recursion tailored for combination with semi-definite approaches, bringing us further down to O ( n 0.19996 ) colors.