Extremal Bounds for Three-Neighbour Bootstrap Percolation in Dimensions Two and Three

The $r$-neighbour bootstrap percolation process on a graph $G$ starts with an initial set $A_0$ of "infected" vertices and, at each step of the process, a healthy vertex becomes infected if it has at least $r$ infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of $G$ eventually becomes infected, then we say that $A_0$ percolates. We prove a conjecture of Balogh and Bollob\'as which says that, for fixed $r$ and $d\to\infty$, ev...

Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G, r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we consider th...

By bootstrap percolation we mean the following deterministic process on a graph $G$. Given a set $A$ of vertices "infected" at time 0, new vertices are subsequently infected, at each time step, if they have at least $r\in\mathbb{N}$ previously infected neighbors. When the set $A$ is chosen at random, the main aim is to determine the critical probability $p_c(G,r)$ at which percolation (infection of the entire graph) becomes likely to occur. This bootstrap process has been ext...

We consider a classic model known as bootstrap percolation on the $n \times n$ square grid. To each vertex of the grid we assign an initial state, infected or healthy, and then in consecutive rounds we infect every healthy vertex that has at least $2$ already infected neighbours. We say that percolation occurs if the whole grid is eventually infected. In this paper, contributing to a recent series of extremal results in this field, we prove that the maximum time a bootstrap p...

The $r$-neighbour bootstrap process is an update rule for the states of vertices in which `uninfected' vertices with at least $r$ `infected' neighbours become infected and a set of initially infected vertices is said to \emph{percolate} if eventually all vertices are infected. For every $r \geq 3$, a sharp condition is given for the minimum degree of a sufficiently large graph that guarantees the existence of a percolating set of size $r$. In the case $r=3$, for $n$ large eno...

In this paper we fill in a fundamental gap in the extremal bootstrap percolation literature, by providing the first proof of the fact that for all $d \geq 1$, the size of the smallest percolating sets in $d$-neighbour bootstrap percolation on $[n]^d$, the $d$-dimensional grid of size $n$, is $n^{d-1}$. Additionally, we prove that such sets percolate in time at most $c_d n^2$, for some constant $c_d >0 $ depending on $d$ only.

In $r$-neighbor bootstrap percolation on the vertex set of a graph $G$, a set $A$ of initially infected vertices spreads by infecting, at each time step, all uninfected vertices with at least $r$ previously infected neighbors. When the elements of $A$ are chosen independently with some probability $p$, it is natural to study the critical probability $p_c(G,r)$ at which it becomes likely that all of $V(G)$ will eventually become infected. Improving a result of Balogh, Bollob\'...

Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^3$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,3\}$, where $a_1\le a_2\le a_3$. Suppose we infect any healthy vertex $v\in [L]^3$ already having $r$ infected neighbours, and that infected sites remain infected forever. In this paper we determine $\log$ of the critical length for percolation u...

In standard bootstrap percolation, a subset A of the n x n grid is initially infected. A new site is then infected if at least two of its neighbours are infected, and an infected site stays infected forever. The set A is said to percolate if eventually the entire grid is infected. A percolating set is said to be minimal if none of its subsets percolate. Answering a question of Bollobas, we show that there exists a minimal percolating set of size 4n^2/33 + o(n^2), but there do...

The $r$-neighbor bootstrap percolation is a graph infection process based on the update rule by which a vertex with $r$ infected neighbors becomes infected. We say that an initial set of infected vertices propagates if all vertices of a graph $G$ are eventually infected, and the minimum cardinality of such a set in $G$ is called the $r$-bootstrap percolation number, $m(G,r)$, of $G$. In this paper, we study percolating sets in direct products of graphs. While in general graph...