TY - GEN

T1 - Coloring k-colorable graphs using smaller palettes

AU - Halperin, Eran

AU - Nathaniel, Ram

AU - Zwick, Uri

PY - 2001

Y1 - 2001

N2 - We obtain the following new coloring results: • A 3-colorable graph on n vertices with maximum degree &Dgr; can be colored, in polynomial time, using &Oor; ((&Dgr; log &Dgr;)1/3 -log n) colors. This slightly improves an &Ogr;((&Dgr;1/3 log1,/2 &Dgr;) log n) bound given by Karger, Motwani and Sudan. More generally, k-colorable graphs with maximum degree &Dgr; can be colored, in polynomial time, using &Ogr;(( &Dgr; 1-2/k; log1/k &Dgr;) log n) colors. A 4-colorable graph on n vertices can be colored, in polynomial time, using &Ogr;(n7/19) colors. This improves an &Ogr;(n2/5) bound given again by Karger, Motwani and Sudan. More generally, k-colorable graphs on n-vertices can be colored, in polynomial time, using &Ogr;(n&agr;k) colors, where &agr;5 = 97/207, &agr;6 = 43/79, &agr;7 = 1391/2315, &agr;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.

AB - We obtain the following new coloring results: • A 3-colorable graph on n vertices with maximum degree &Dgr; can be colored, in polynomial time, using &Oor; ((&Dgr; log &Dgr;)1/3 -log n) colors. This slightly improves an &Ogr;((&Dgr;1/3 log1,/2 &Dgr;) log n) bound given by Karger, Motwani and Sudan. More generally, k-colorable graphs with maximum degree &Dgr; can be colored, in polynomial time, using &Ogr;(( &Dgr; 1-2/k; log1/k &Dgr;) log n) colors. A 4-colorable graph on n vertices can be colored, in polynomial time, using &Ogr;(n7/19) colors. This improves an &Ogr;(n2/5) bound given again by Karger, Motwani and Sudan. More generally, k-colorable graphs on n-vertices can be colored, in polynomial time, using &Ogr;(n&agr;k) colors, where &agr;5 = 97/207, &agr;6 = 43/79, &agr;7 = 1391/2315, &agr;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.

KW - Algorithms

KW - Theory

KW - Verification

UR - http://www.scopus.com/inward/record.url?scp=35448990454&partnerID=8YFLogxK

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:35448990454

SN - 0898714907

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 319

EP - 326

BT - Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms

T2 - 2001 Operating Section Proceedings, American Gas Association

Y2 - 30 April 2001 through 1 May 2001

ER -