The concentration of the chromatic number of random graphs


We prove that for every constant δ>0 the chromatic number of the random graphG(n, p) withp=n −1/2−δ is asymptotically almost surely concentrated in two consecutive values. This implies that for any β<1/2 and any integer valued functionr(n)O(n β) there exists a functionp(n) such that the chromatic number ofG(n,p(n)) is preciselyr(n) asymptotically almost surely.

This is a preview of subscription content, access via your institution.


  1. [1]

    N. Alon: Restricted colorings of graphs, in:Surveys in Combinatorics, Proc. 14th British Combinatorial Conference, London Mathematical Society Lecture Notes Series 187, K. Walker ed., Cambridge University Press, 1993, 1–33.

  2. [2]

    N. Alon andJ. H. Spencer:The probabilistic method, Wiley, New York, 1992.

    Google Scholar 

  3. [3]

    B. Bollobás: The chromatic number of random graphs,Combinatorica,8 (1988), 49–55.

    Google Scholar 

  4. [4]

    A. Frieze andB. Reed: Covering the edges of a random graph by cliques,Combinatorica,15 (1995), 489–497.

    Google Scholar 

  5. [5]

    T. R. Jensen andB. Toft:Graph coloring problems, Wiley, New York, 1995.

    Google Scholar 

  6. [6]

    M. Krivelevich: On the minimal number of edges in color-critical graphs,Combinatorica,17 (1997), 401–426.

    Google Scholar 

  7. [7]

    T. Luczak: The chromatic number of random graphs,Combinatorica,11 (1991), 45–54.

    Google Scholar 

  8. [8]

    T. Łuczak: A note on the sharp concentration of the chromatic number of random graphs.Combinatorica,11 (1991), 295–297.

    Google Scholar 

  9. [9]

    D. W. Matula: On the complete subgraph of a random graph,Combinatory Mathematics and its Applications, Chapel Hill, North Carolina (1970), 356–369.

    Google Scholar 

  10. [10]

    E. Shamir andJ. Spencer: Sharp concentration of the chromatic number of random graphsG n,p ,Combinatorica,7 (1987), 124–129.

    Google Scholar 

Download references

Author information



Additional information

Research supported in part by a USA Israeli BSF grant and by a grant from the Israel Science Foundation.

Research supported in part by a Charles Clore Fellowship.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Alon, N., Krivelevich, M. The concentration of the chromatic number of random graphs. Combinatorica 17, 303–313 (1997).

Download citation

Mathematics Subject Classification (1991)

  • 05C80
  • 05C15