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# Graphs with tiny vector chromatic numbers and huge chromatic numbers

Feige, Uriel and Langberg, Michael and Schechtman, Gideon (2004) Graphs with tiny vector chromatic numbers and huge chromatic numbers. SIAM Journal on Computing, 33 (6). pp. 1338-1368. ISSN 0097-5397. http://resolver.caltech.edu/CaltechAUTHORS:20111012-111737057

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## Abstract

Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246-265] introduced the notion of a vector coloring of a graph. In particular, they showed that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly Δ^(1 - 2/k) colors. Here Δ is the maximum degree in the graph and is assumed to be of the order of n^5 for some 0 < δ < 1. Their results play a major role in the best approximation algorithms used for coloring and for maximum independent sets. We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than n/Δ^(1- 2/k) (and hence cannot be colored with significantly fewer than Δ^(1-2/k) colors). For k = O(log n/log log n) we show vector k-colorable graphs that do not have independent sets of size (log n)^c, for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylogn. As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653-750] for this problem.

Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1137/S0097539703431391 DOIUNSPECIFIED
http://epubs.siam.org/sicomp/resource/1/smjcat/v33/i6/p1338_s1PublisherUNSPECIFIED
Additional Information:© 2004 Society for Industrial and Applied Mathematics. Received by the editors July 9, 2003; accepted for publication (in revised form) March 2, 2004; published electronically August 6, 2004. We would like to thank Luca Trevisan for his suggestion to analyze edge sampling. This author was supported in part by the Israel Science Foundation (grant 236/02). This author’s work was done while studying at the Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel 76100. The work of this author was supported in part by the Israel Science Foundation (grant 154/01).
Funders:
Funding AgencyGrant Number
Israel Science Foundation236/02
Israeli Science Foundation154/01
Subject Keywords:semidefinite programming, chromatic number, independent set, approximation algorithms, property testing
Classification Code:AMS subject classifications: 68R05, 05C15, 90C22
Record Number:CaltechAUTHORS:20111012-111737057
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20111012-111737057
Official Citation:Graphs with Tiny Vector Chromatic Numbers and Huge Chromatic Numbers Uriel Feige, Michael Langberg, and Gideon Schechtman SIAM J. Comput. 33, pp. 1338-1368
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:27189
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:13 Oct 2011 14:52