Coloring k-colorable graphs using relatively small palettes
release_6xkb4yuosjdm7avwdaj5j7jb74
by
Eran Halperin, Ram Nathaniel, Uri Zwick
2001
Abstract
We obtain the following new coloring results:
* A 3-colorable graph on n vertices with maximum degree Δ can be
colored, in polynomial time, using O((ΔΔ)^1/3·n)
colors. This slightly improves an O((Δ^1/3^1/2Δ)·n) bound given by Karger, Motwani and Sudan. More
generally, k-colorable graphs with maximum degree Δ can be colored, in
polynomial time, using O((Δ^1-2/k^1/kΔ) ·n)
colors.
* A 4-colorable graph on n vertices can be colored, in polynomial time,
using (n^7/19) colors. This improves an (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 (n^α_k) colors,
where α_5=97/207, α_6=43/79, α_7=1391/2315,
α_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.
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