## 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 A is the maximum degree in the graph and is assumed to be of the order of n ^{2} for some 0 < 6 < 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 (logn) ^{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.

Original language | English |
---|---|

Pages (from-to) | 1338-1368 |

Number of pages | 31 |

Journal | SIAM Journal on Computing |

Volume | 33 |

Issue number | 6 |

DOIs | |

State | Published - 2004 |

Externally published | Yes |

### Bibliographical note

Funding Information:This paper was supported by Konkuk University in 2000.

## Keywords

- Approximation algorithms
- Chromatic number
- Independent set
- Property testing
- Semidefinite programming