Given a graph G and p is an element of [0, 1], the random subgraph G(p) is obtained by retaining each edge of G independently with probability p. We show that for every is an element of > 0, there exists a constant C > 0 such that the following holds. Let d >= C be an integer, let G be a d-regular graph and let p >= C/d . Then, with probability tending to one as |V(G)| tends to infinity, there exists a matching in G(p) covering at least (1- is an element of)|V(G)| vertices. We further show that for a wide family of d-regular graphs G, which includes the d-dimensional hypercube, for any p >= log(5 )d/d with probability tending to one as d tends to infinity, G(p) contains an induced subgraph on at least (1-o(1)|)V (G)| vertices, whose degrees are tightly concentrated around the expected average degree dp.

We provide a sufficient condition on the isoperimetric properties of a regular graph $G$ of growing degree $d$, under which the random subgraph $G_p$ typically undergoes a phase transition around $p=\frac{1}{d}$ which resembles the emergence of a giant component in the binomial random graph model $G(n,p)$. We further show that this condition is tight. More precisely, let $d=\omega(1)$, let $\epsilon>0$ be a small enough constant, and let $p \cdot d=1+\epsilon$. We show that if $C$ is sufficiently large and $G$ is a $d$-regular $n$-vertex graph where every subset $S\subseteq V(G)$ of order at most $\frac{n}{2}$ has edge-boundary of size at least $C|S|$, then $G_p$ typically has a unique linear sized component, whose order is asymptotically $y(\epsilon)n$, where $y(\epsilon)$ is the survival probability of a Galton-Watson tree with offspring distribution Po$(1+\epsilon)$. We further give examples to show that this result is tight both in terms of its dependence on $C$, and with respect to the order of the second-largest component. We also consider a more general setting, where we only control the expansion of sets up to size $k$. In this case, we show that if $G$ is such that every subset $S\subseteq V(G)$ of order at most $k$ has edge-boundary of size at least $d|S|$ and $p$ is such that $p\cdot d \geq 1 + \epsilon$, then $G_p$ typically contains a component of order $\Omega(k)$.

Let $d\ge 3$ be a fixed integer. Let $y:= y(p)$ be the probability that the root of an infinite $d$-regular tree belongs to an infinite cluster after $p$-bond-percolation. We show that for all constants $b,\alpha>0$ and $1<\lambda < d-1$, there exist constants $c,C>0$ such that the following holds. Let $G$ be a $d$-regular graph on $n$ vertices, satisfying that for every $U\subseteq V(G)$ with $|U|\le \frac{n}{2}$, $e(U,U^{c})\ge b|U|$ and for every $U\subseteq V(G)$ with $|U|\le \log ^{C}n$, $e(U)\le (1+c)|U|$. Let $p=\frac{\lambda }{d-1}$. Then, with probability tending to one as $n$ tends to infinity, the largest component $L_{1}$ in the random subgraph $G_{p}$ of $G$ satisfies $\left |1-\frac{|L_{1}|}{yn}\right |\le \alpha $, and all the other components in $G_{p}$ are of order $O\left (\frac{\lambda \log n}{(\lambda -1)^{2}}\right )$. This generalises (and improves upon) results for random $d$-regular graphs.

Let $K<^>r_n$ be the complete $r$ -uniform hypergraph on $n$ vertices, that is, the hypergraph whose vertex set is $[n] \, :\! = \{1,2,\ldots ,n\}$ and whose edge set is $\binom {[n]}{r}$ . We form $G<^>r(n,p)$ by retaining each edge of $K<^>r_n$ independently with probability $p$ . An $r$ -uniform hypergraph $H\subseteq G$ is $F$ -saturated if $H$ does not contain any copy of $F$ , but any missing edge of $H$ in $G$ creates a copy of $F$ . Furthermore, we say that $H$ is weakly $F$ -saturated in $G$ if $H$ does not contain any copy of $F$ , but the missing edges of $H$ in $G$ can be added back one-by-one, in some order, such that every edge creates a new copy of $F$ . The smallest number of edges in an $F$ -saturated hypergraph in $G$ is denoted by ${\textit {sat}}(G,F)$ , and in a weakly $F$ -saturated hypergraph in $G$ by $\mathop {\mbox{$w$-${sat}$}}\! (G,F)$ . In 2017, Kor & aacute;ndi and Sudakov initiated the study of saturation in random graphs, showing that for constant $p$ , with high probability ${\textit {sat}}(G(n,p),K_s)=(1+o(1))n\log _{\frac {1}{1-p}}n$ , and $\mathop {\mbox{$w$-${sat}$}}\! (G(n,p),K_s)=\mathop {\mbox{$w$-${sat}$}}\! (K_n,K_s)$ . Generalising their results, in this paper, we solve the saturation problem for random hypergraphs $G<^>r(n,p)$ for cliques $K_s<^>r$ , for every $2\le r \lt s$ and constant $p$ .

Given a graph $G$ and probability $p$, we form the random subgraph $G_p$ by retaining each edge of $G$ independently with probability $p$. Given $d\in\mathbb{N}$, constants $00$, and $p=\frac{1+\varepsilon}{d}$, we show that if every subset $S\subseteq V(G)$ of size exactly $\frac{c|V(G)|}{d}$ satisfies $|N(S)|\ge d|S|$, then the probability that $G_p$ does not contain a cycle of length $\Omega(\varepsilon^2c^2|V(G)|)$ is exponentially small in $|V(G)|$. As an intermediate step, we also show that given $k,d\in \mathbb{N}$, a constant $\varepsilon>0$, and $p=\frac{1+\varepsilon}{d}$, if every subset $S\subseteq V(G)$ of size exactly $k$ satisfies $|N(S)|\ge kd$, then the probability that $G_p$ does not contain a path of length $\Omega(\varepsilon^2 kd)$ is exponentially small. We further discuss applications of these results to $K_{s,t}$-free graphs of maximal density.

Let d,n is an element of N be such that d = omega(1), and d <= n(1-a) for some constant a > 0. Consider a d-regular graph G = (V, E) and the random graph process that starts with the empty graph G (0) and at each step G(i) is obtained from G(i-1) by adding uniformly at random a new edge from E. We show that if G satisfies some (very) mild global edge-expansion, and an almost optimal edge-expansion of sets up to order O(d log n), then for any constant k is an element of N in the random graph process on G, typically the hitting times of minimum degree at least k and of k-connectedness are equal. This, in particular, covers both d-regular high dimensional product graphs and pseudo-random graphs, and confirms a conjecture of Joos from 2015. We further demonstrate that this result is tight in the sense that there are d-regular n-vertex graphs with optimal edge-expansion of sets up to order Omega(d), for which the probability threshold of minimum degree at least one is different than the probability threshold of connectivity. (c) 2025 The Author(s). Published by Elsevier B.V.

For graphs $G$ and $F$, the saturation number $\textit{sat}(G,F)$ is the minimum number of edges in an inclusion-maximal $F$-free subgraph of $G$. In 2017, Kor\'andi and Sudakov initiated the study of saturation in random graphs. They showed that for constant $p\in (0,1)$, whp $\textit{sat}\left(G(n,p),K_s\right)=\left(1+o(1)\right)n\log_{\frac{1}{1-p}}n$. We show that for every graph $F$ and every constant $p\in (0,1)$, whp $\textit{sat}\left(G(n,p), F\right)=O(n\ln n)$. Furthermore, if every edge of $F$ belongs to a triangle, then the above is the right asymptotic order of magnitude, that is, whp $\textit{sat}\left(G(n,p),F\right)=\Theta(n\ln n)$. We further show that for a large family of graphs $\mathcal{F}$ with an edge that does not belong to a triangle, which includes all the bipartite graphs, for every $F\in \mathcal{F}$ and constant $p\in(0,1)$, whp $\textit{sat}\left(G(n,p),F\right)=O(n)$. We conjecture that this sharp transition from $O(n)$ to $\Theta(n\ln n)$ depends only on this property, that is, that for any graph $F$ with at least one edge that does not belong to a triangle, whp $\textit{sat}\left(G(n,p),F\right)=O(n)$. We further generalise the result of Kor\'andi and Sudakov, and show that for a more general family of graphs $\mathcal{F}'$, including all complete graphs $K_s$ and all complete multipartite graphs of the form $K_{1,1,s_3,\ldots, s_{\ell}}$, for every $F\in \mathcal{F}'$ and every constant $p\in(0,1)$, whp $\textit{sat}\left(G(n,p),F\right)=\left(1+o(1)\right)n\log_{\frac{1}{1-p}}n$. Finally, we show that for every complete multipartite graph $K_{s_1, s_2, \ldots, s_{\ell}}$ and every $p\in \left[\frac{1}{2},1\right)$, $\textit{sat}\left(G(n,p),K_{s_1,s_2,\ldots,s_{\ell}}\right)=\left(1+o(1)\right)n\log_{\frac{1}{1-p}}n$.

We consider bond percolation on high-dimensional product graphs $G=\square _{i=1}^tG^{(i)}$ , where $\square$ denotes the Cartesian product. We call the $G^{(i)}$ the base graphs and the product graph $G$ the host graph. Very recently, Lichev ( J. Graph Theory , 99(4):651–670, 2022) showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph $G_p$ undergoes a phase transition when $p$ is around $\frac{1}{d}$ , where $d$ is the average degree of the host graph. In the supercritical regime, we strengthen Lichev’s result by showing that the giant component is in fact unique, with all other components of order $o(|G|)$ , and determining the sharp asymptotic order of the giant. Furthermore, we answer two questions posed by Lichev ( J. Graph Theory , 99(4):651–670, 2022): firstly, we provide a construction showing that the requirement of bounded degree is necessary for the likely emergence of a linear order component; secondly, we show that the isoperimetric requirement on the base graphs can be, in fact, super-exponentially small in the dimension. Finally, in the subcritical regime, we give an example showing that in the case of irregular high-dimensional product graphs, there can be a polynomially large component with high probability, very much unlike the quantitative behaviour seen in the Erdős-Rényi random graph and in the percolated hypercube, and in fact in any regular high-dimensional product graphs, as shown by the authors in a companion paper (Percolation on high-dimensional product graphs. arXiv:2209.03722 , 2022).

We provide sufficient conditions for a regular graph G of growing degree d, guaranteeing a phase transition in its random subgraph G_p similar to that of G(n,p) when p· d≈ 1. These conditions capture several well-studied graphs, such as (percolation on) the complete graph K_n, the binary hypercube Q^d, d-regular expanders, and random d-regular graphs. In particular, this serves as a unified proof for these (and other) cases. Suppose that G is a d-regular graph on n vertices, with d=ω(1). Let ε>0 be a small constant, and let p=1+ε/d. Let y(ε) be the survival probability of a Galton-Watson tree with offspring distribution Po(1+ε). We show that if G satisfies a (very) mild edge expansion requirement, and if one has fairly good control on the expansion of small sets in G, then typically the percolated random subgraph G_p contains a unique giant component of asymptotic order y(ε)n, and all the other components in G_p are of order O(log n/ε^2). We also show that this result is tight, in the sense that if one asks for a slightly weaker control on the expansion of small sets in G, then there are d-regular graphs G on n vertices, where typically the second largest component is of order Ω(dlog (n/d))=ω(log n). This is the first of a two-part sequence of papers. In the subsequent work, we consider supercritical percolation on regular graphs of constant degree, and establish similar sufficient (and essentially tight) conditions in that setting.

A classical vertex Ramsey result due to Nešetřil and Rödl states that given a finite family of graphs F, a graph A and a positive integer r, if every graph B∈F has a 2-vertex-connected subgraph which is not a subgraph of A, then there exists an F-free graph which is vertex r-Ramsey with respect to A. We prove that this sufficient condition for the existence of an F-free graph which is vertex r-Ramsey with respect to A is also necessary for large enough number of colours r. We further show a generalisation of the result to a family of graphs and the typical existence of such a subgraph in a dense binomial random graph.

We present a short and self-contained proof of a classical result due to Bollobás (1990): in the random hypercube process, with high probability the hitting time of connectedness equals the hitting time of having minimum degree at least one.

Let $Q^d$ be the $d$-dimensional binary hypercube. We say that $P=\{v_1,\ldots, v_k\}$ is an increasing path of length $k-1$ in $Q^d$, if for every $i\in [k-1]$ the edge $v_iv_{i+1}$ is obtained by switching some zero coordinate in $v_i$ to a one coordinate in $v_{i+1}$. Form a random subgraph $Q^d_p$ by retaining each edge in $E(Q^d)$ independently with probability $p$. We show that there is a phase transition with respect to the length of a longest increasing path around $p=\frac{e}{d}$. Let $\alpha$ be a constant and let $p=\frac{\alpha}{d}$. When $\alphae$, whp there is a path of length $d-2$ in $Q^d_p$, and in fact, whether it is of length $d-2, d-1$, or $d$ depends on whether the all-zero and all-one vertices percolate or not.

It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a classic result for the hypercube. In this paper we give new isoperimetric inequalities for such regular high-dimensional product graphs, which generalise the well-known isoperimetric inequality of Harper for the hypercube, and are asymptotically sharp for a wide range of set sizes. We then use these isoperimetric properties to investigate the structure of the giant component L_1 in supercritical percolation on these product graphs, that is, when p=1+ϵ/d , where d is the degree of the product graph and ϵ >0 is a small enough constant. We show that typically L_1 has edge-expansion Ω( 1/dln d) . Furthermore, we show that L_1 likely contains a linear-sized subgraph with vertex-expansion Ω( 1/dln d) . These results are best possible up to the logarithmic factor in d. Using these likely expansion properties, we determine, up to small polylogarithmic factors in d, the likely diameter of L_1 as well as the typical mixing time of a lazy random walk on L_1 . Furthermore, we show the likely existence of a cycle of length Ω( n/dln d) . These results not only generalise, but also improve substantially upon the known bounds in the case of the hypercube, where in particular the likely diameter and typical mixing time of L_1 were previously only known to be polynomial in d.

For t ∈ℕ and every i∈[t], let H_i be a d_i-regular connected graph, with 1<|V(H_i)|≤ C for some integer C≥ 2. Let G=□_i=1^tH_i be the Cartesian product of H_1, …, H_t. We show that if t≥ 5C then G contains a (nearly-)perfect matching. Then, considering the random graph process on G, we generalise the result of Bollobás on the binary hypercube Q^t, showing that with high probability, the hitting times for minimum degree one, connectivity, and the existence of a (nearly-)perfect matching in the random graph process on G are the same. As a byproduct, we develop several tools which may be of independent interest in a more general setting when one seeks to establish the typical existence of a perfect matching under percolation.

In 2004, Frieze, Krivelevich and Martin established the emergence of a giant component in random subgraphs of pseudo‐random graphs. We study several typical properties of the giant component, most notably its expansion characteristics. We establish an asymptotic vertex expansion of connected sets in the giant by a factor of . From these expansion properties, we derive that the diameter of the giant is whp , and that the mixing time of a lazy random walk on the giant is asymptotically . We also show similar asymptotic expansion properties of (not necessarily connected) linear‐sized subsets in the giant, and the typical existence of a large expander as a subgraph.

We consider supercritical site percolation on the $d$ -dimensional hypercube $Q^d$ . We show that typically all components in the percolated hypercube, besides the giant, are of size $O(d)$ . This resolves a conjecture of Bollobás, Kohayakawa, and Łuczak from 1994.

We establish the existence of the phase transition in site percolation on pseudo-random $d$-regular graphs. Let $G=(V,E)$ be an $(n,d,\lambda)$-graph, that is, a $d$-regular graph on $n$ vertices in which all eigenvalues of the adjacency matrix, but the first one, are at most $\lambda$ in their absolute values. Form a random subset $R$ of $V$ by putting every vertex $v\in V$ into $R$ independently with probability $p$. Then for any small enough constant $\epsilon>0$, if $p=\frac{1-\epsilon}{d}$, then with high probability all connected components of the subgraph of $G$ induced by $R$ are of size at most logarithmic in $n$, while for $p=\frac{1+\epsilon}{d}$, if the eigenvalue ratio $\lambda/d$ is small enough as a function of $\epsilon$, then typically $R$ contains a connected component of size at least $\frac{\epsilon n}{d}$ and a path of length proportional to $\frac{\epsilon^2n}{d}$.

Given a graph $G$, we denote by $f(G,u_0,k)$ the number of paths of length $k$ in $G$ starting from $u_0$. In graphs of maximum degree 3, with edge weights $i.i.d.$ with $exp(1)$, we provide a simple proof showing that (under the assumption that $f(G,u_0,k)=\omega(1)$) the expected weight of the heaviest path of length $k$ in $G$ starting from $u_0$ is at least \begin{align*} (1-o(1))\left(k+\frac{\log_2\left(f(G,u_0,k)\right)}{2}\right), \end{align*} and the expected weight of the lightest path of length $k$ in $G$ starting from $u_0$ is at most \begin{align*} (1+o(1))\left(k-\frac{\log_2\left(f(G,u_0,k)\right)}{2}\right). \end{align*} We demonstrate the immediate implication of this result for Hamilton paths and Hamilton cycles in random cubic graphs, where we show that typically there exist paths and cycles of such weight as well. Finally, we discuss the connection of this result to the question of a longest cycle in the giant component of supercritical $G(n,p)$.

We consider the performance of the Depth First Search (DFS) algorithm on the random graph $G\left(n,\frac{1+\epsilon}{n}\right)$, $\epsilon>0$ a small constant. Recently, Enriquez, Faraud and Ménard proved that the stack $U$ of the DFS follows a specific scaling limit, reaching the maximal height of $\left(1+o_{\epsilon}(1)\right)\epsilon^2n$. Here we provide a simple analysis for the typical length of a maximum path discovered by the DFS.