We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

A graph $G$ is Ramsey for a graph $H$ if every 2-colouring of the edges of $G$ contains a monochromatic copy of $H$. We consider the following question: if $H$ has bounded treewidth, is there a `sparse' graph $G$ that is Ramsey for $H$? Two notions of sparsity are considered. Firstly, we show that if the maximum degree and treewidth of $H$ are bounded, then there is a graph $G$ with $O(|V(H)|)$ edges that is Ramsey for $H$. This was previously only known for the smaller class of graphs $H$ with bounded bandwidth. On the other hand, we prove that the treewidth of a graph $G$ that is Ramsey for $H$ cannot be bounded in terms of the treewidth of $H$ alone. In fact, the latter statement is true even if the treewidth is replaced by the degeneracy and $H$ is a tree.

The size Ramsey number of graphs with bounded treewidth

We develop a general embedding method based on the Friedman-Pippenger tree embedding technique (1987) and its algorithmic version, essentially due to Aggarwal et al. (1996), enhanced with a roll-back idea allowing to sequentially retrace previously performed embedding steps. This proves to be a powerful tool for embedding graphs of large girth into expander graphs. As an application of this method, we settle two problems: 
-For a graph $H$, we denote by $H^q$ the graph obtained from $H$ by subdividing its edges with $q{-}1$ vertices each. We show that the $k$-size-Ramsey number $\hat{R}_k(H^q)$ satisfies $\hat{R}_k(H^q)=O(qn)$ for every bounded degree graph $H$ on $n$ vertices and for $q=\Omega(\log n)$, which is optimal up to a constant factor. This settles a conjecture of Pak (2002). 
-We give a deterministic, polynomial time algorithm for finding vertex-disjoint paths between given pairs of vertices in a strong expander graph. More precisely, let $G$ be an $(n,d,\lambda)$-graph with $\lambda=O(d^{1-\varepsilon})$, and let $\mathcal{P}$ be any collection of at most $c\frac{ n\log d}{\log n}$ disjoint pairs of vertices in $G$ for some small constant $c$, such that in the neighborhood of every vertex in $G$ there are at most $d/4$ vertices from $\mathcal{P}$. Then there exists a polynomial time algorithm which finds vertex-disjoint paths between every pair in $\mathcal{P}$, and each path is of the same length $\ell=O\left(\frac{\log n}{\log d}\right)$. Both the number of pairs and the length of the paths are optimal up to a constant factor; the result answers the offline version of a question of Alon and Capalbo (2007).

Rolling backwards can move you forward: on embedding problems in sparse expanders

We study Hamiltonicity in random subgraphs of the hypercube $\mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $\mathcal{Q}^n$ according to a uniformly chosen random ordering. Then, with high probability, as soon as the graph produced by this process has minimum degree $2k$, it contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. Secondly, we obtain a perturbation result: if $H\subseteq\mathcal{Q}^n$ satisfies $\delta(H)\geq\alpha n$ with $\alpha>0$ fixed and we consider a random binomial subgraph $\mathcal{Q}^n_p$ of $\mathcal{Q}^n$ with $p\in(0,1]$ fixed, then with high probability $H\cup\mathcal{Q}^n_p$ contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. In particular, both results resolve a long standing conjecture, posed e.g. by Bollobas, that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals $1/2$. Our techniques also show that, with high probability, for all fixed $p\in(0,1]$ the graph $\mathcal{Q}^n_p$ contains an almost spanning cycle. Our methods involve branching processes, the Rodl nibble, and absorption.

Hamiltonicity of random subgraphs of the hypercube

Given a positive integer $s$, the $s$-colour size-Ramsey number of a graph $H$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that, in any colouring of $E(G)$ with $s$ colours, there is a monochromatic copy of $H$. We prove that, for any positive integers $k$ and $s$, the $s$-colour size-Ramsey number of the $k$th power of any $n$-vertex bounded degree tree is linear in $n$. As a corollary we obtain that the $s$-colour size-Ramsey number of $n$-vertex graphs with bounded treewidth and bounded degree is linear in $n$, which answers a question raised by Kamcev, Liebenau, Wood and Yepremyan [The size Ramsey number of graphs with bounded treewidth, arXiv:1906.09185 (2019)].

The size-Ramsey number of powers of bounded degree trees

In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_\alpha$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac's theorem and the results by Posa and Korsunov on the threshold in $\mathbb{G}(n,p)$. In this note we extend this result in $\mathcal{G}_\alpha \cup \mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$. More precisely, for any $\varepsilon>0$ and $\alpha \colon \mathbb{N} \mapsto (0,1)$ we show that a.a.s. $\mathcal{G}_\alpha \cup \mathbb{G}(n,\beta /n)$ is Hamiltonian, where $\beta = -(6 + \varepsilon) \log(\alpha)$. If $\alpha>0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $\alpha=O(1/n)$ the random part $\mathbb{G}(n,p)$ is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into $\mathbb{G}(n,p)$.

Random Perturbation of Sparse Graphs

We consider sufficient conditions for the existence of $k$-th powers of Hamiltonian cycles in $n$-vertex graphs $G$ with minimum degree $\mu n$ for arbitrarily small $\mu>0$. About 20 years ago Komlos, Sarkozy, and Szemeredi resolved the conjectures of Posa and Seymour and obtained optimal minimum degree conditions for this problem by showing that $\mu=\frac{k}{k+1}$ suffices for large $n$. For smaller values of $\mu$ the given graph $G$ must satisfy additional assumptions. We show that inducing subgraphs of density $d>0$ on linear subsets of vertices and being inseparable, in the sense that every cut has density at least $\mu>0$, are sufficient assumptions for this problem and, in fact, for a variant of the bandwidth theorem. This generalises recent results of Staden and Treglown.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/rsa.20957

Giulia Satiko Maesaka

Papers

The size-Ramsey number of powers of bounded degree trees

Random Perturbation of Sparse Graphs

The size-Ramsey number of powers of bounded degree trees

Embedding spanning subgraphs in uniformly dense and inseparable graphs

Random perturbation of sparse graphs