The Lovasz number of random graphs

Abstract

We study the Lovasz number theta along with two further SDP relaxations theta1, theta1/2 of the independence number and the corresponding relaxations of the chromatic number on random graphs G(n,p). We prove that these relaxations are concentrated about their means Moreover, extending a result of Juhasz, we compute the asymptotic value of the relaxations for essentially the entire range of edge probabilities p. As an application, we give an improved algorithm for approximating the independence number in polynomial expected time, thereby extending a result of Krivelevich and Vu. We also improve on the analysis of an algorithm of Krivelevich for deciding whether G(n,p) is k-colorable.