Randomized graph products, chromatic numbers, and the Lovász ϑ-function

AbstractFor a graphG, let α(G) denote the size of the largest independent set inG, and let ϑ(G) denote the Lovász ϑ-function onG. We prove that for somec>0, there exists an infinite family of graphs such that $$\vartheta (G) > \alpha (G)n/2^{c\sqrt {\log n} }$$ , wheren denotes the number of vertices in a graph. this disproves a known conjecture regarding the ϑ function.As part of our proof, we analyse the behavior of the chromatic number in graphs under a randomized version of graph products. This analysis extends earlier work of Linial and Vazirani, and of Berman and Schnitger, and may be of independent interest.

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