Coloring k-colorable graphs using relatively small palettes

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) . log n) colors. This slightly improves an O((Delta(1/3) log(1/2) Delta) . 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) . log n) colors. A 4-colorable graph on n vertices can be colored, in polynomial time, using (O) over tilde (n(7/19)) colors. This improves an (O) over tilde (n(2/5)) bound given again by Karaer. Motwani, and Sudan. More generally, k-colorable graphs on n vertices can be colored, in polynomial time, using (O) over tilde (n(alphak)) 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 relined 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. (C) 2002 Elsevier Science (USA). All rights reserved.