On the concentration of the chromatic number of random graphs

TL;DR

This paper improves the understanding of how the chromatic number of random graphs concentrates, showing tighter bounds in sparse cases and a surprising jump in concentration in very dense graphs.

Contribution

It extends previous concentration results to sparse graphs and reveals a new phenomenon of concentration jump in dense graphs.

Findings

Improved concentration bounds for sparse random graphs.

Identification of a concentration jump in very dense graphs.

Extension of classical results to new regimes.

Abstract

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 \omega\sqrt{n}, and in the 1990s Alon showed that an interval of length \omega\sqrt{n}/\log n suffices for constant edge-probabilities p \in (0,1). We prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case p=p(n) \to 0, and uncover a surprising concentration `jump' of the chromatic number in the very dense case p=p(n) \to 1.