On the concentration of the chromatic number of random graphs
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Shamir and Spencer proved in the 1980s that the chromatic number of the binomial random graph G(n,p) is concentrated in an interval of length at most ω√(n), and in the 1990s Alon showed that an interval of length ω√(n)/logn suffices for constant edge-probabilities p ∈(0,1). We prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case p=p(n) →0, and uncover a surprising concentration `jump' of the chromatic number in the very dense case p=p(n) →1.
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