On the concentration of the chromatic number of random graphs

TL;DR

This paper discusses the concentration of the chromatic number in random graphs, improving the known bounds from about n^{1/2} to about n^{1/2}/log n, based on a proof by Noga Alon.

Contribution

It provides a proof that the chromatic number of G(n,p) is concentrated in a narrower interval of length about n^{1/2}/log n, refining previous results.

Findings

Chromatic number concentration interval is about n^{1/2}/log n.

Improves upon previous concentration bounds of about n^{1/2}.

Based on a proof by Noga Alon.

Abstract

Let 0<p<1 be fixed. Shamir and Spencer proved in the 1980s that the chromatic number of a random graph in G(n,p) is concentrated in an interval of length about n^{1/2}. In this explanatory note, we give a proof of a result due due Noga Alon, showing that the chromatic number is concentrated in an interval of length about n^{1/2}/log n.