, Volume 8, Issue 1, pp 49–55 | Cite as

The chromatic number of random graphs

  • B. Bollobás


For a fixed probabilityp, 0<p<1, almost every random graphGn,p has chromatic number
$$\left( {\frac{1}{2} + o(1)} \right)\log (1/(1 - p))\frac{n}{{\log n}}$$

AMS subject classification (1980)

05 C 15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    K. Azuma, Weighted sums of certain dependent random variables,Tôhoku Math. J.,3 (1967), 357–367.Google Scholar
  2. [2]
    B. Bollobás, The evolution of sparse graphs,in Graph Theory and Combinatorics, Proc. Cambridge Combinatorial Conf. in honour of Paul Erdős (B. Bollobás, ed.), Academic Press, London, 1984, 35–37.Google Scholar
  3. [3]
    B. Bollobás,Random Graphs, Academic Press, London, 1985, xvi+447 pp.Google Scholar
  4. [4]
    B. Bollobás andP. Erdős, Cliques in random graphs,Math. Proc. Cambridge Phil. Soc.,80 (1976), 419–427.Google Scholar
  5. [5]
    B.Bollobás and A. G.Thomason, Random graphs of small order,in Random Graphs, Annals of Discr. Math., 1985, 47–97.Google Scholar
  6. [6]
    P. Erdős andJ. Spencer,Probabilistic Methods in Combinatorics, Academic Press, New York and London, 1974.Google Scholar
  7. [7]
    D. Freedman, On tail probabilities for martingales,Ann. Probab.,3 (1975), 100–118.Google Scholar
  8. [8]
    G. R. Grimmett andC. J. H. McDiarmid, On colouring random graphs,Math. Proc. Cambridge Phil. Soc.,77 (1975), 313–324.Google Scholar
  9. [9]
    W. B. Johnson andG. Schechtman, Embeddingl pm intol 1n,Acta Math.,150 (1983), 71–85.Google Scholar
  10. [10]
    C. J. H.McDiarmid, Colouring random graphs badly,in Graph Theory and Combinatorics (R. J. Wilson, ed.), Pitman Research Notes in Mathematics,34 (1979), 76–86.Google Scholar
  11. [11]
    C. J. H. McDiarmid, On the chromatic forcing number of a random graph,Discrete Appl. Math.,5 (1983), 123–132.Google Scholar
  12. [12]
    D. W. Matula, Expose-and-merge exploration and the chromatic number of a random graph,Combinatorica,7 (1987), 275–284.Google Scholar
  13. [13]
    B. Maurey, Construction de suites symétriques,Compt. Rend. Acad. Sci. Paris 288 (1979), 679–681.Google Scholar
  14. [14]
    V. D.Milman and G.Schechtman,Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, 1986, viii+156 pp.Google Scholar
  15. [15]
    G. Pisier, On the dimension ofl pn-subspaces of Banach spaces, for 1<p<2,Trans. Amer. Math. Soc.,276 (1983), 201–211.Google Scholar
  16. [16]
    E. Shamir andJ. Spencer, Sharp concentration of the chromatic number of random graphsG n,p,Combinatorica,7 (1987), 121–129.Google Scholar
  17. [17]
    G. Schechtman, Random embeddings of euclidean spaces in sequence spaces,Israel J. Math.,40 (1981), 187–192.Google Scholar
  18. [18]
    E. Shamir andR. Upfal, Sequential and distributed graph colouring algorithms with performance analysis in random graph spaces,J. of Algorithms 5 (1984), 488–501.Google Scholar
  19. [19]
    W. F. De La Vega, On the chromatic number of sparse random graphs,in Graph Theory and Combinatorics, Proc. Cambridge Combinatorial Conf. in honour of Paul Erdős (B. Bollobás, ed., Academic Press, London, 1984, 321–328.Google Scholar

Copyright information

© Akadémiai Kiadó 1988

Authors and Affiliations

  • B. Bollobás
    • 1
  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeEngland

Personalised recommendations