Coloring k-colorable graphs using relatively small palettes
Eran Halperin, Ram Nathaniel, Uri Zwick
备注：
16 pages, 2 figures. A preliminary version of this paper appeared in the proceedings the of 12th ACM-SIAM Symposium on Discrete Algorithm (SODA'01) under a slightly different title
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 $O((Δ\logΔ)^{1/3} \cdot\log{n})$ colors. This slightly improves an $O((Δ^{{1}/{3}} \log^{1/2}Δ)\cdot\log{n})$ bound given by Karger, Motwani and Sudan. More generally, $k$-colorable graphs with maximum degree $Δ$ can be colored, in polynomial time, using $O((Δ^{1-{2}/{k}}\log^{1/k}Δ) \cdot\log{n})$ 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 $\Ot(n^{α_k})$ 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.
Improved SDP Coloring of 3-Colorable Graphs from Recursive Gaussian Certificates
Georgia Institute of Technology, School of Computer Science（佐治亚理工学院计算机学院）；Princeton University, Operations Research and Financial Engineering（普林斯顿大学运筹与金融工程学系）
Approximate Graph Coloring by Semidefinite Programming
We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on $n$ vertices with min O(Delta^{1/3} log^{1/2} Delta log n), O(n^{1/4} log^{1/2} n) colors where Delta is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of n, this marks the first non-trivial approximation result as a function of the maximum degree Delta. This result can be generalized to k-colorable graphs to obtain a coloring using min O(Delta^{1-2/k} log^{1/2} Delta log n), O(…
