Combinatorics, Probability and Computing


The Lovász Number of Random Graphs

a1 Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, 10099 Berlin, Germany (e-mail:

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coja-oghlan a   [Google Scholar] 


We study the Lovász number $\vartheta$ along with two related SDP relaxations $\vartheta_{1/2}$, $\vartheta_2$ of the independence number and the corresponding relaxations $\bar\vartheta$, $\bar\vartheta_{1/2}$, $\bar\vartheta_2$ of the chromatic number on random graphs $G_{n,p}$. We prove that $\vartheta,\vartheta_{1/2},\vartheta_2(G_{n,p})$ are concentrated about their means, and that $\bar\vartheta,\bar\vartheta_{1/2},\bar\vartheta_2(G_{n,p})$ in the case $p<n^{-1/2-\varepsilon}$ are concentrated in intervals of constant length. Moreover, extending a result of Juhász [28], we estimate the probable value of $\vartheta,\vartheta_{1/2},\vartheta_2(G_{n,p})$ for edge probabilities $c_0/n\leq p\leq 1-c_0/n$, where $c_0>0$ is a constant. As an application, we give improved algorithms for approximating the independence number of $G_{n,p}$ and for deciding $k$-colourability in polynomial expected time.

(Received June 19 2003)
(Revised December 22 2003)


1 Research supported by the Deutsche Forschungsgemeinschaft (grant DFG FOR 413/1-1).