We study the Lovász number ϑ along with two further SDP relaxations ϑ 1/2, ϑ 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 \(\bar\vartheta,\bar\vartheta_{1/2},\bar\vartheta_2(G_{n,p})\) in the case p<n  − 1/2 −  ε are concentrated in intervals of constant length. Moreover, we estimate the probable value of \(\vartheta,\bar\vartheta(G_{n,p})\) etc. for essentially the entire range of edge probabilities p. As applications, we give improved algorithms for approximating α(G n, p ) and for deciding k-colorability in polynomial expected time.


Random Graph Chromatic Number Vertex Cover Input Graph Independence Number 
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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  1. 1.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany

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