Coloring k-colorable graphs using relatively small palettes

Abstract

We obtain the following new coloring results: * A 3-colorable graph on $n$ vertices with maximum degree~$\Delta$ can be colored, in polynomial time, using $O((\Delta \log\Delta)^{1/3} \cdot\log{n})$ colors. This slightly improves an $O((\Delta^{{1}/{3}} \log^{1/2}\Delta)\cdot\log{n})$ bound given by Karger, Motwani and Sudan. More generally, $k$-colorable graphs with maximum degree $\Delta$ can be colored, in polynomial time, using $O((\Delta^{1-{2}/{k}}\log^{1/k}\Delta) \cdot\log{n})$ colors. * A 4-colorable graph on $n$ vertices can be colored, in polynomial time, using $\Ot(n^{7/19})$ colors. This improves an $\Ot(n^{2/5})$ bound given again by Karger, Motwani and Sudan. More generally, $k$-colorable graphs on $n$ vertices can be colored, in polynomial time, using $\Ot(n^{\alpha_k})$ colors, where $\alpha_5=97/207$, $\alpha_6=43/79$, $\alpha_7=1391/2315$, $\alpha_8=175/271$, ... The first result is obtained by a slightly more refined probabilistic analysis of the semidefinite programming based coloring algorithm of Karger, Motwani and Sudan. The second result is obtained by combining the coloring algorithm of Karger, Motwani and Sudan, the combinatorial coloring algorithms of Blum and an extension of a technique of Alon and Kahale (which is based on the Karger, Motwani and Sudan algorithm) for finding relatively large independent sets in graphs that are guaranteed to have very large independent sets. The extension of the Alon and Kahale result may be of independent interest.