Related papers: Strong-majority bootstrap percolation on regular g…

A bootstrap percolation process on a graph with infection threshold $r\ge 1$ is a dissemination process that evolves in time steps. The process begins with a subset of infected vertices and in each subsequent step every uninfected vertex…

Consider a graph $G=(V,E)$ without isolated edges and with maximum degree $\Delta$. Given a colouring $c:E\to\{1,2,\ldots,k\}$, the weighted degree of a vertex $v\in V$ is the sum of its incident colours, i.e., $\sum_{e\ni v}c(e)$. For any…

We show that for all $d\in \{3,\ldots,n-1\}$ the size of the largest component of a random $d$-regular graph on $n$ vertices around the percolation threshold $p=1/(d-1)$ is $\Theta(n^{2/3})$, with high probability. This extends known…

In this paper we study percolation on a roughly transitive graph G with polynomial growth and isoperimetric dimension larger than one. For these graphs we are able to prove that p_c < 1, or in other words, that there exists a percolation…

The classical Erd\H{o}s-P\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k)…

We study the evolution of majority dynamics on Erd\H{o}s-R\'enyi $G(n,p)$ random graphs. In this process, each vertex of a graph is assigned one of two initial states. Subsequently, on every day, each vertex simultaneously updates its state…

In this paper, we study the k-neighbor bootstrap percolation process on the d-dimensional grid [n]^d, and show that the minimum number of initial vertices that percolate is (1-d/k)n^d + O(n^{d-1})$ when d<=k<=2d. This confirms a conjecture…

Let $G$ be a vertex-transitive graph of superlinear polynomial growth. Given $r>0$, let $G_r$ be the graph on the same vertex set as $G$, with two vertices joined by an edge if and only if they are at graph distance at most $r$ apart in…

Given two graphs $G$ and $H$, it is said that $G$ percolates in $H$-bootstrap process if one could join all the nonadjacent pairs of vertices of $G$ in some order such that a new copy of $H$ is created at each step. Balogh, Bollob\'as and…

We present a rigorous and precise analysis of the maximum degree and the average degree in a dynamic duplication-divergence graph model introduced by Sol\'e, Pastor-Satorras et al. in which the graph grows according to a…

We investigate the dynamic formation of regular random graphs. In our model, we pick a pair of nodes at random and connect them with a link if both of their degrees are smaller than d. Starting with a set of isolated nodes, we repeat this…

In this paper, we study the following problem. Consider a setting where a proposal is offered to the vertices of a given network $G$, and the vertices must conduct a vote and decide whether to accept the proposal or reject it. Each vertex…

For Bernoulli percolation on a given graph $G = (V,E)$ we consider the cluster of some fixed vertex $o \in V$. We aim at comparing the number of vertices of this cluster in the set $V_+$ and in the set $V_-$, where $V_+,V_- \subset V$ have…

We consider the Constrained-degree percolation model in random environment on the square lattice. In this model, each vertex $v$ has an independent random constraint ${\kappa}_v$ which takes the value $j\in \{0,1,2,3\}$ with probability…

Equip each point $x$ of a homogeneous Poisson process $\mathcal{P}$ on $\mathbb{R}$ with $D_x$ edge stubs, where the $D_x$ are i.i.d. positive integer-valued random variables with distribution given by $\mu$. Following the stable…

Graph bootstrap percolation, introduced by Bollob\'as in 1968, is a cellular automaton defined as follows. Given a "small" graph $H$ and a "large" graph $G = G_0 \subseteq K_n$, in consecutive steps we obtain $G_{t+1}$ from $G_t$ by adding…

Order the vertices of a directed random graph \math{v_1,...,v_n}; edge \math{(v_i,v_j)} for \math{i<j} exists independently with probability \math{p}. This random graph model is related to certain spreading processes on networks. We…

For a constant $\gamma \in[0,1]$ and a graph $G$, let $\omega_{\gamma}(G)$ be the largest integer $k$ for which there exists a $k$-vertex subgraph of $G$ with at least $\gamma\binom{k}{2}$ edges. We show that if $0<p<\gamma<1$ then…

A set $D$ of vertices is a strong dominating set in a graph $G$, if for every vertex $x\in V(G) \setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x) \leq deg(y)$. The strong domination number $\gamma_{st}(G)$ of $G$ is the…

By a theorem of Volkov (2001) we know that on most graphs with positive probability the linearly vertex-reinforced random walk (VRRW) stays within a finite "trapping" subgraph at all large times. The question of whether this tail behavior…