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 Õ(n
^4/11
) colors. This is the first combinatorial improvement of Blum's Õ(n
^3/8
) bound from FOCS'90. Like Blum's algorithm, our new algorithm composes nicely with recent semi-definite programming approaches. The current best bound is Õ(n
^0.2072
) colors by Chlamtac from FOCS'07. We now bring it down to Õ(n
^{0. 2049}
) colors.