Colouring random graphs: Tame colourings

TL;DR

This paper investigates the concentration properties of t-bounded colourings in random graphs, revealing two-point concentration results and their implications for the distribution of the chromatic number.

Contribution

It introduces the concept of tame colourings and provides tight probabilistic bounds, advancing understanding of colourings in random graphs and supporting the Zigzag Conjecture.

Findings

hi_t(G(n,m)) is maximally concentrated on at most two explicit values for t=lpha(G(n,m))-2

hi_t(G_{n,1/2}) is contained in a length n^{0.99} interval with high probability under certain conditions

Proves two-point concentration of the equitable chromatic number of G(n,m)

Abstract

Given a graph G, a colouring is an assignment of colours to the vertices of G so that no two adjacent vertices are coloured the same. If all colour classes have size at most t, then we call the colouring t-bounded, and the t-bounded chromatic number of G, denoted by $\chi_t(G)$, is the minimum number of colours in such a colouring. Every colouring of G is then $\alpha(G)$-bounded, where $\alpha(G)$ denotes the size of a largest independent set. We study colourings of the random graph G(n, 1/2) and of the corresponding uniform random graph G(n,m) with $m=\left \lfloor \frac 12 {n \choose 2} \right \rfloor$. We show that $\chi_t(G(n,m))$ is maximally concentrated on at most two explicit values for $t = \alpha(G(n,m))-2$. This behaviour stands in stark contrast to that of the normal chromatic number, which was recently shown not to be concentrated on any sequence of intervals of length $n^{1/2-o(1)}$. Moreover, when $t = \alpha(G_{n, 1/2})-1$ and if the expected number of independent sets of size $t$ is not too small, we determine an explicit interval of length $n^{0.99}$ that contains $\chi_t(G_{n,1/2})$ with high probability. Both results have profound consequences: the former is at the core of the intriguing Zigzag Conjecture on the distribution of $\chi(G_{n, 1/2})$ and justifies one of its main hypotheses, while the latter is an important ingredient in the proof of a non-concentration result for $\chi(G_{n,1/2})$ that is conjectured to be optimal. These two results are consequences of a more general statement. We consider a class of colourings that we call tame, and provide tight bounds for the probability of existence of such colourings via a delicate second moment argument. We then apply those bounds to the two aforementioned cases. As a further consequence of our main result, we prove two-point concentration of the equitable chromatic number of G(n,m).