Coloring 3-Colorable Graphs with Less than n1/5 Colors.

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(n4/11) colors. This is the first combinatorial improvement since Blum’s Õ(n3/8) 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(n0.2072) colors by Chlamtac from FOCS’07 to O(n0.2049) colors. Next, we develop a new recursion tailored for combination with semi-definite approaches, bringing us further down to O(n0.19996) colors.