Related papers: A Sharp Threshold for Bootstrap Percolation in a R…

An elegant bootstrap percolation result on the hyperrectangle graph was proved by Balogh, Bollob\'as, Morris, and Riordan using a linear algebric method in 2012. This paper studies the same problem by purely combinatorial means. The base…

We determine the sharp threshold for Hamilton cycles in randomly perturbed sparse graphs. For any $\alpha=\alpha(n)=o(1)$, let $G_{\alpha}$ be an $n$-vertex graph with minimum degree $\delta(G_{\alpha})\ge\alpha n$. We prove that if…

As introduced by Bollob\'as, a graph $G$ is weakly $H$-saturated if the complete graph $K_n$ is obtained by iteratively completing copies of $H$ minus an edge. For all graphs $H$, we obtain an asymptotic lower bound for the critical…

A $k$-uniform hypergraph $H = (V, E)$ is called $\ell$-orientable, if there is an assignment of each edge $e\in E$ to one of its vertices $v\in e$ such that no vertex is assigned more than $\ell$ edges. Let $H_{n,m,k}$ be a hypergraph,…

We give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for…

We study the contact process on a dynamic random~$d$-regular graph with an edge-switching mechanism, as well as an interacting particle system that arises from the local description of this process, called the herds process. Both these…

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed…

In this paper we consider a simple virus infection spread model on a finite population of $n$ agents connected by some neighborhood structure. Given a graph $G$ on $n$ vertices, we begin with some fixed number of initial infected vertices.…

We study phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a…

An irreversible $k$-threshold process (also a $k$-neighbor bootstrap percolation) is a dynamic process on a graph where vertices change color from white to black if they have at least $k$ black neighbors. An irreversible $k$-conversion set…

In the polluted bootstrap percolation model, vertices of the cubic lattice $\mathbb{Z}^3$ are independently declared initially occupied with probability $p$ or closed with probability $q$. Under the standard (respectively, modified)…

We consider bootstrap percolation and diffusion in sparse random graphs with fixed degrees, constructed by configuration model. Every node has two states: it is either active or inactive. We assume that to each node is assigned a…

We establish the sharpness of the percolation phase transition for a class of infinite-range weighted random connection models. The vertex set is given by a marked Poisson point process on $\mathbb{R}^d$ with intensity $\lambda>0$, where…

Let $d\geq 3$ be a constant and let $F$ be a $d$-regular graph on $[n]$ with not too many symmetries. By the union bound, the probability threshold for the existence of a spanning subgraph in $G(n,p)$ isomorphic to $F$ is at least…

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes (PPPs) in $\bbR ^2$ of intensities $\lambda$ and $\lambda_E$…

Percolation is a model for random damage to a network. It is one of the simplest models that displays a phase transition: when the network is severely damaged, it falls apart in many small connected components, while if the damage is light,…

We investigate locality of the supercritical regime for Bernoulli percolation on transitive graphs with polynomial growth, by which we mean the following. Take a transitive graph of polynomial growth $\mathscr{G}$ satisfying…

K-core and bootstrap percolation are widely studied models that have been used to represent and understand diverse deactivation and activation processes in natural and social systems. Since these models are considerably similar, it has been…

Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of a graph a random number is assigned and a path through the graph is called accessible if all numbers along the path are in…

Let $H$ be a triple system with maximum degree $d>1$ and let $r>10^7\sqrt{d}\log^{2}d$. Then $H$ has a proper vertex coloring with $r$ colors such that any two color classes differ in size by at most one. The bound on $r$ is sharp in order…