How does the chromatic number of a random graph vary?

The chromatic number χ(G) of a graph G is a fundamental parameter, whose study was originally motivated by applications (χ(G) is the minimum number of internally compatible groups the vertices can be divided into, if the edges represent incompatibility). As with other graph parameters, it is also studied from a purely theoretical point of view, and here a key question is: what is its typical value? More precisely, how does χ(Gn,1/2), 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 √ n, improved slightly by Alon to √ 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 Bollob´as. 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 √ n log log n/ log3 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(Gn,1/2) is, and why. The first form of this conjecture arises from recent work of Bollob´as, 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

Colouring random graphs: Tame colourings

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).Comment: 75 page

Combinatorics

Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session

Combinatorics, Probability and Computing

The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. The workshop also emphasized connections between probabilistic combinatorics and discrete probability

How sharp is the concentration of the chromatic number?

The chromatic number of a random graph was determined. If the probability is small then the concentration of the chromatic number is even sharper. These concentration results did not improve at all the known bounds in the chromatic number. The results show that chromatic number is very sharply concentrated about a known value