The Lovász Number of Random Graphs
Combinatorics, Probability and Computing
New approximation guarantee for chromatic number
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Approximating Maximum Subgraphs without Short Cycles
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
CSP gaps and reductions in the lasserre hierarchy
Proceedings of the forty-first annual ACM symposium on Theory of computing
Approximating Maximum Subgraphs without Short Cycles
SIAM Journal on Discrete Mathematics
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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 $\Delta^{1 - 2/k}$ colors. Here $\Delta$ is the maximum degree in the graph and is assumed to be of the order of $n^{\delta}$ for some $0 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 $n/\Delta^{1 - 2/k}$ (and hence cannot be colored with significantly fewer than $\Delta^{1 - 2/k}$ 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/polylog n. 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.