Let ${\mathbb{G}(n_1,n_2,m)}$ be a uniformly random m-edge subgraph of the complete bipartite graph ${K_{n_1,n_2}}$ with bipartition $(V_1, V_2)$ , where $n_i = |V_i|$ , $i=1,2$ . Given a real number $p \in [0,1]$ such that $d_1 \,{:\!=}\, pn_2$ and $d_2 \,{:\!=}\, pn_1$ are integers, let $\mathbb{R}(n_1,n_2,p)$ be a random subgraph of ${K_{n_1,n_2}}$ with every vertex $v \in V_i$ of degree $d_i$ , $i = 1, 2$ . In this paper we determine sufficient conditions on $n_1,n_2,p$ and m under which one can embed ${\mathbb{G}(n_1,n_2,m)}$ into $\mathbb{R}(n_1,n_2,p)$ and vice versa with probability tending to 1. In particular, in the balanced case $n_1=n_2$ , we show that if $p\gg\log n/n$ and $1 - p \gg \left(\log n/n \right)^{1/4}$ , then for some $m\sim pn^2$ , asymptotically almost surely one can embed ${\mathbb{G}(n_1,n_2,m)}$ into $\mathbb{R}(n_1,n_2,p)$ , while for $p\gg\left(\log^{3} n/n\right)^{1/4}$ and $1-p\gg\log n/n$ the opposite embedding holds. As an extension, we confirm the Kim–Vu Sandwich Conjecture for degrees growing faster than $(n \log n)^{3/4}$ .

A conjecture of Alon, Krivelevich and Sudakov states that, for any graph $F$ , there is a constant $c_F \gt 0$ such that if $G$ is an $F$ -free graph of maximum degree $\Delta$ , then $\chi\!(G) \leqslant c_F \Delta/ \log\!\Delta$ . Alon, Krivelevich and Sudakov verified this conjecture for a class of graphs $F$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot and Sereni that if $G$ is $K_{t,t}$ -free, then $\chi\!(G) \leqslant (t + o(1)) \Delta/ \log\!\Delta$ as $\Delta \to \infty$ . We improve this bound to $(1+o(1)) \Delta/\log\!\Delta$ , making the constant factor independent of $t$ . We further extend our result to the DP-colouring setting (also known as correspondence colouring), introduced by Dvořák and Postle.

We study several parameters of a random Bienaymé–Galton–Watson tree $T_n$ of size $n$ defined in terms of an offspring distribution $\xi$ with mean $1$ and nonzero finite variance $\sigma ^2$ . Let $f(s)=\mathbb{E}\{s^\xi \}$ be the generating function of the random variable $\xi$ . We show that the independence number is in probability asymptotic to $qn$ , where $q$ is the unique solution to $q = f(1-q)$ . One of the many algorithms for finding the largest independent set of nodes uses a notion of repeated peeling away of all leaves and their parents. The number of rounds of peeling is shown to be in probability asymptotic to $\log n/\log (1/f'(1-q))$ . Finally, we study a related parameter which we call the leaf-height. Also sometimes called the protection number, this is the maximal shortest path length between any node and a leaf in its subtree. If $p_1 = \mathbb{P}\{\xi =1\}\gt 0$ , then we show that the maximum leaf-height over all nodes in $T_n$ is in probability asymptotic to $\log n/\log (1/p_1)$ . If $p_1 = 0$ and $\kappa$ is the first integer $i\gt 1$ with $\mathbb{P}\{\xi =i\}\gt 0$ , then the leaf-height is in probability asymptotic to $\log _\kappa \log n$ .

We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$ -vertex graph $G$ satisfying a given minimum degree condition and the binomial random graph $G(n,p)$ . We prove that asymptotically almost surely $G \cup G(n,p)$ contains at least $\min \{\delta (G), \lfloor n/3 \rfloor \}$ pairwise vertex-disjoint triangles, provided $p \ge C \log n/n$ , where $C$ is a large enough constant. This is a perturbed version of an old result of Dirac.

Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516] in a strong form. We also prove a stability version of our result, which in the case of pairwise vertex-disjoint triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159–176], this fully resolves the existence of triangle factors in randomly perturbed graphs.

We believe that the methods introduced in this paper are useful for a variety of related problems: we discuss possible generalisations to clique factors, cycle factors, and $2$ -universality.

The clustered chromatic number of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$ -colourable with monochromatic components of size at most $c$ . We determine the clustered chromatic number of any minor-closed class with bounded treedepth, and prove a best possible upper bound on the clustered chromatic number of any minor-closed class with bounded pathwidth. As a consequence, we determine the fractional clustered chromatic number of every minor-closed class.

Given a graph $H$ and a positive integer $n$ , the Turán number $\mathrm{ex}(n,H)$ is the maximum number of edges in an $n$ -vertex graph that does not contain $H$ as a subgraph. A real number $r\in (1,2)$ is called a Turán exponent if there exists a bipartite graph $H$ such that $\mathrm{ex}(n,H)=\Theta (n^r)$ . A long-standing conjecture of Erdős and Simonovits states that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q\gt p$ .

In this paper, we show that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q \gt p^{2}$ . Our result also addresses a conjecture of Janzer [18].

Let $G=(S,T,E)$ be a bipartite graph. For a matching $M$ of $G$ , let $V(M)$ be the set of vertices covered by $M$ , and let $B(M)$ be the symmetric difference of $V(M)$ and $S$ . We prove that if $M$ is a uniform random matching of $G$ , then $B(M)$ satisfies the BK inequality for increasing events.

We show that the $4$ -state anti-ferromagnetic Potts model with interaction parameter $w\in (0,1)$ on the infinite $(d+1)$ -regular tree has a unique Gibbs measure if $w\geq 1-\dfrac{4}{d+1_{_{\;}}}$ for all $d\geq 4$ . This is tight since it is known that there are multiple Gibbs measures when $0\leq w\lt 1-\dfrac{4}{d+1}$ and $d\geq 4$ . We moreover give a new proof of the uniqueness of the Gibbs measure for the $3$ -state Potts model on the $(d+1)$ -regular tree for $w\geq 1-\dfrac{3}{d+1}$ when $d\geq 3$ and for $w\in (0,1)$ when $d=2$ .