The Chromatic Number of Random Regular Graphs

  • Dimitris Achlioptas
  • Cristopher Moore
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3122)

Abstract

Given any integer d ≥ 3, let k be the smallest integer such that d < 2k log k. We prove that with high probability the chromatic number of a random d-regular graph is k, k+1, or k+2.

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References

  1. 1.
    Achlioptas, D., Moore, C.: Almost all graphs of degree 4 are 3-colorable. Journal of Computer and System Sciences 67(4), 441–471 (2003)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Achlioptas, D., Moore, C.: The asymptotic order of the k-SAT threshold. In: Proc. 43rd Foundations of Computer Science, pp. 779–788 (2002)Google Scholar
  3. 3.
    Achlioptas, D., Moore, C.: On the two-colorability of random hypergraphs. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 78–90. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Achlioptas, D., Naor, A.: The two possible values of the chromatic number of a random graph. In: Proc. 36th Symp. on the Theory of Computing (2004)Google Scholar
  5. 5.
    Bollobás, B.: Random graphs. Academic Press, London (1985)MATHGoogle Scholar
  6. 6.
    Frieze, A., Luczak, T.: On the independence and chromatic numbers of random regular graphs. J. Combin. Theory Ser. B 54, 123–132 (1992)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Janson, S., Lucza, T., Ruciński, A.: Random Graphs. Wiley & Sons, Chichester (2000)MATHGoogle Scholar
  8. 8.
    Krza̧ka_la, F., Pagnani, A., Weigt, M.: Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs. Preprint, cond-mat/0403725. Physical Review (to appear) Google Scholar
  9. 9.
    Luczak, T.: The chromatic number of random graphs. Combinatorica 11(1), 45–54 (1991)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Luczak, T.: A note on the sharp concentration of the chromatic number of random graphs. Combinatorica 11(3), 295–297 (1991)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Molloy, M.: The Chromatic Number of Sparse Random Graphs. Master’s thesis, Faculty of Mathematics, University of Waterloo (1992) Google Scholar
  12. 12.
    Wormald, N.C.: Models of random regular graphs. In: Lamb, J.D., Preece, D.A. (eds.) Surveys in Combinatorics., London Mathematical Society Lecture Note Series, vol. 276, pp. 239–298. Cambridge University Press, Cambridge (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Dimitris Achlioptas
    • 1
  • Cristopher Moore
    • 2
  1. 1.Microsoft ResearchRedmondUSA
  2. 2.University of New MexicoUSA

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