Maximal induced paths and minimal percolating sets in hypercubes

For a graph $G$, the \emph{$r$-bootstrap percolation} process can be described as follows: Start with an initial set $A$ of "infected'' vertices. Infect any vertex with at least $r$ infected neighbours, and continue this process until no new vertices can be infected. $A$ is said to \emph{percolate in $G$} if eventually all the vertices of $G$ are infected. $A$ is a \emph{minimal percolating set} in $G$ if $A$ percolates in $G$ and no proper subset of $A$ percolates in $G$. An induced path, $P$, in a hypercube $Q_n$ is maximal if no induced path in $Q_n$ properly contains $P$. Induced paths in hypercubes are also called snakes. We study the relationship between maximal snakes and minimal percolating sets (under 2-bootstrap percolation) in hypercubes. In particular, we show that every maximal snake contains a minimal percolating set, and that every minimal percolating set is contained in a maximal snake

Maximal Bootstrap Percolation Time on the Hypercube via Generalised Snake-in-the-Box

In $r$-neighbour bootstrap percolation, vertices (sites) of a graph $G$ are infected, round-by-round, if they have $r$ neighbours already infected. Once infected, they remain infected. An initial set of infected sites is said to percolate if every site is eventually infected. We determine the maximal percolation time for $r$-neighbour bootstrap percolation on the hypercube for all $r \geq 3$ as the dimension $d$ goes to infinity up to a logarithmic factor. Surprisingly, it turns out to be $\frac{2^d}{d}$, which is in great contrast with the value for $r=2$, which is quadratic in $d$, as established by Przykucki. Furthermore, we discover a link between this problem and a generalisation of the well-known Snake-in-the-Box problem.Comment: 14 pages, 1 figure, submitte

Line percolation

We study a new geometric bootstrap percolation model, line percolation, on the $d$-dimensional integer grid $[n]^d$. In line percolation with infection parameter $r$, infection spreads from a subset $A\subset [n]^d$ of initially infected lattice points as follows: if there exists an axis-parallel line $L$ with $r$ or more infected lattice points on it, then every lattice point of $[n]^d$ on $L$ gets infected, and we repeat this until the infection can no longer spread. The elements of the set $A$ are usually chosen independently, with some density $p$, and the main question is to determine $p_c(n,r,d)$, the density at which percolation (infection of the entire grid) becomes likely. In this paper, we determine $p_c(n,r,2)$ up to a multiplicative factor of $1+o(1)$ and $p_c(n,r,3)$ up to a multiplicative constant as $n\rightarrow \infty$ for every fixed $r\in \mathbb{N}$. We also determine the size of the minimal percolating sets in all dimensions and for all values of the infection parameter.Comment: 27 pages, Random Structures and Algorithm