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Sahar Diskin

Publications and source records attributed to Sahar Diskin.

32 records · Page 2Linked to original sources
arXiv2024

Long cycles in percolated expanders

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

math.CO↗
arXiv2024

Hitting time of connectedness in the random hypercube process

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.

math.PR↗
arXiv2024

Percolation on High-dimensional Product Graphs

We consider percolation on high-dimensional product graphs, where the base graphs are regular and of bounded order. In the subcritical regime, we show that typically the largest component is of order logarithmic in the number of vertices. In the supercritical regime, our main result recovers the sharp asymptotic of the order of the largest component, and shows that all the other components are typically of order logarithmic in the number of vertices. In particular, we show that this phase transition is quantitatively similar to the one of the binomial random graph. This generalises the results of Ajtai, Komlós, and Szemerédi and of Bollobás, Kohayakawa, and Łuczak who showed that the $d$-dimensional hypercube, which is the $d$-fold Cartesian product of an edge, undergoes a phase transition quantitatively similar to the one of the binomial random graph.

math.CO↗
arXiv2024

A Jump of the Saturation Number in Random Graphs?

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, Korándi 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 $\textit{sat}\left(G(n,p),F\right)=Θ(n\ln n)$. 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 $Θ(n\ln n)$ 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 Korándi 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 $p\in \left[\frac{1}{2},1\right)$, $\textit{sat}\left(G(n,p),K_{s_1,s_2,\ldots,s_{\ell}}\right)=\left(1+o(1)\right)n\log_{\frac{1}{1-p}}n$.

math.CO↗
arXiv2024

Percolation on Irregular High-dimensional Product Graphs

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 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: 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.

math.CO↗
arXiv2024

Isoperimetric Inequalities and Supercritical Percolation on High-dimensional Graphs

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 $p=\frac{1+ε}{d}$, 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 $Ω\left(\frac{1}{d\ln d}\right)$. Furthermore, we show that $L_1$ likely contains a linear-sized subgraph with vertex-expansion $Ω\left(\frac{1}{d\ln d}\right)$. 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 path of length $Ω\left(\frac{n}{d\ln d}\right)$. 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 $d$.

math.PR↗
arXiv2024

Percolation through Isoperimetry

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 $d=ω(1)$, let $ε>0$ be a small enough constant, and let $p \cdot d=1+ε$. 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 $y(ε)n$, where $y(ε)$ is the survival probability of a Galton-Watson tree with offspring distribution Po$(1+ε)$. 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 $p\cdot d \geq 1 + ε$, then $G_p$ typically contains a component of order $Ω(k)$.

math.CO↗
arXiv2023

Climbing up a random subgraph of the hypercube

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 $α$ be a constant and let $p=\fracα{d}$. When $α e$, 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.

math.CO↗
arXiv2023

On vertex Ramsey graphs with forbidden subgraphs

A classical vertex Ramsey result due to Nešetřil and Rödl states that given a finite family of graphs $\mathcal{F}$, a graph $A$ and a positive integer $r$, if every graph $B\in\mathcal{F}$ has a $2$-vertex-connected subgraph which is not a subgraph of $A$, then there exists an $\mathcal{F}$-free graph which is vertex $r$-Ramsey with respect to $A$. We prove that this sufficient condition for the existence of an $\mathcal{F}$-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.

math.CO↗
arXiv2023

Heavy and Light Paths and Hamilton Cycles

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 $f(G,u_0,k)=ω(1)$) 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 $G(n,p)$.

math.CO↗
arXiv2022

Site Percolation on Pseudo-Random Graphs

We consider vertex percolation on pseudo-random $d-$regular graphs. The previous study by the second author established the existence of phase transition from small components to a linear (in $\frac{n}{d}$) sized component, at $p=\frac{1}{d}$. In the supercritical regime, our main result recovers the sharp asymptotic of the size of the largest component, and shows that all other components are typically much smaller. Furthermore, we consider other typical properties of the largest component such as the number of edges, existence of a long cycle and expansion. In the subcritical regime, we strengthen the upper bound on the likely component size.

math.CO↗
arXiv2022

On the Performance of the Depth First Search Algorithm in Supercritical Random Graphs

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

math.CO↗
arXiv2022

Expansion in Supercritical Random Subgraphs of Expanders and its Consequences

In 2004, Frieze, Krivelevich and Martin [17] 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 $\tilde{O}\left(ε^2\right)$. From these expansion properties, we derive that the diameter of the giant is typically $O_ε\left(\log n\right)$, and that the mixing time of a lazy random walk on the giant is asymptotically $O_ε\left(\log^2 n\right)$. 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.

math.CO↗