How does the chromatic number of a random graph vary?

Abstract

How does the chromatic number of a graph chosen uniformly at random from all graphs on $n$ vertices behave? This quantity is a random variable, so one can ask (i) for upper and lower bounds on its typical values, and (ii) for bounds on how much it varies: what is the width (e.g., standard deviation) of its distribution? On (i) there has been considerable progress over the last 45 years; on (ii), which is our focus here, remarkably little. One would like both upper and lower bounds on the width of the distribution, and ideally a description of the (appropriately scaled) limiting distribution. There is a well known upper bound of Shamir and Spencer of order $\sqrt{n}$, improved slightly by Alon to $\sqrt{n}/\log n$, but no non-trivial lower bound was known until 2019, when the first author proved that the width is at least $n^{1/4-o(1)}$ for infinitely many $n$, answering a longstanding question of Bollobás. In this paper we have two main aims: first, we shall prove a much stronger lower bound on the width. We shall show unconditionally that, for some values of $n$, the width is at least $n^{1/2-o(1)}$, matching the upper bounds up to the error term. Moreover, conditional on a recently announced sharper explicit estimate for the chromatic number, we improve the lower bound to order $\sqrt{n} \log \log n /\log^3 n$, within a logarithmic factor of the upper bound. Secondly, we will describe a number of conjectures as to what the true behaviour of the variation in $χ(G_{n,1/2})$ is, and why. The first form of this conjecture arises from recent work of Bollobás, Heckel, Morris, Panagiotou, Riordan and Smith. We will also give much more detailed conjectures, suggesting that the true width, for the worst case $n$, matches our lower bound up to a constant factor. These conjectures also predict a Gaussian limiting distribution.