The concentration of the chromatic number of random graphs

Abstract

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.

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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.

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Alon, N., Krivelevich, M. The concentration of the chromatic number of random graphs. Combinatorica 17, 303–313 (1997). https://doi.org/10.1007/BF01215914

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Mathematics Subject Classification (1991)

  • 05C80
  • 05C15