Colouring graphs whose chromatic number is almost their maximum degree

  • Michael Molloy
  • Bruce Reed
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1380)


We present efficient algorithms for determining if the chromatic number of an input graph is close to δ. Our results are obtained via the probabilistic method.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. Alon and J. Spencer, The Probabilistic Method. Wiley (1992).Google Scholar
  2. 2.
    K. Azuma, Weighted sums of certain dependent random variables. Tokuku Math. Journal 19 (1967), 357–367.zbMATHMathSciNetGoogle Scholar
  3. 3.
    R.L. Brooks, On colouring the nodes of a network, Proc. Cambridge Phil. Soc. 37 (1941), 194–197.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    T. Emden-Weinert, S. Hougardy, and B. Kreuter, Uniquely colourable graphs and the hardness of colouring graphs with large girth, Probability, Combinatorics, and Computing, to appear.Google Scholar
  5. 5.
    P. Erdós and L. Lovász, Problems and results on 3-chromatic hypergraphs and some related questions, in: “Infinite and Finite Sets” (A. Hajnal et. al. Eds), Colloq. Math. Soc. J. Bolyai 11, North Holland, Amsterdam, 1975, 609–627.Google Scholar
  6. 6.
    R. Karp, Reducibility among combinatorial problems, In Complexity of Computer Computations, Plenum Press (1972), 85–103.Google Scholar
  7. 7.
    F. Maffray and M. Preissmann, On the NP-completeness of the k-colourability problem for triangle-free graphs, Discrete Mathematics 162 (1996), 313–317.CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    M. Molloy and B. Reed, A bound on the strong chromatic index of a graph, Journal of Combinatorial Theory (B), to appear.Google Scholar
  9. 9.
    M. Molloy and B. Reed, A bound on the total chromatic number, submitted.Google Scholar
  10. 10.
    M. Molloy and B. Reed, Assymptotically better list colourings, manuscript.Google Scholar
  11. 11.
    M. Molloy and B. Reed, An algorithmic version of the Lovasz Local Lemma, in preparation.Google Scholar
  12. 12.
    M. Molloy and B. Reed, Graph Colouring via the Probabilistic Method, to appear in a book edited by A. Gyarfas and L. Lovasz.Google Scholar
  13. 13.
    B. Reed, χ, δ and Ω, Journal of Graph Theory, to appear.Google Scholar
  14. 14.
    B. Reed, A strengthening of Brook's Theorem, in preparation.Google Scholar
  15. 15.
    M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces. Institut Des Hautes études Scientifiques, Publications Mathématiques 81 (1995), 73–205.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Michael Molloy
    • 1
  • Bruce Reed
    • 2
    • 3
  1. 1.Dept. of Computer ScienceUniversity of TorontoTorontoCanada
  2. 2.CNRSParisFrance
  3. 3.IMEUSPSÃo PauloBrazil

Personalised recommendations