Combinatorial coloring of 3-colorable graphs

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 $\text{tO}(n^{4\mathrm{/}11})$ colors. This is the first combinatorial improvement of Blum’s $\text{tO}(n^{3\mathrm{/}8})$ 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.