Saturation in the Hypercube and Bootstrap Percolation

TL;DR

This paper investigates saturation and weak saturation in hypercubes and grids, establishing minimum edge counts for these properties and answering open questions in bootstrap percolation and graph saturation.

Contribution

It provides exact asymptotic bounds for the minimum edges in saturated and weakly saturated hypercube subgraphs, solving open problems posed by Johnson and Pinto.

Findings

Minimum edges in (Q_d,Q_m)-saturated graphs are Θ(2^d).

Exact minimum edges for weakly (Q_d,Q_m)-saturated graphs are determined.

Results extend to grid graphs and cycles, broadening understanding of bootstrap percolation.

Abstract

Let $Q_d$ denote the hypercube of dimension $d$. Given $d\geq m$, a spanning subgraph $G$ of $Q_d$ is said to be $(Q_d,Q_m)$-saturated if it does not contain $Q_m$ as a subgraph but adding any edge of $E(Q_d)\setminus E(G)$ creates a copy of $Q_m$ in $G$. Answering a question of Johnson and Pinto, we show that for every fixed $m\geq2$ the minimum number of edges in a $(Q_d,Q_m)$-saturated graph is $\Theta(2^d)$. We also study weak saturation, which is a form of bootstrap percolation. A spanning subgraph of $Q_d$ is said to be weakly $(Q_d,Q_m)$-saturated if the edges of $E(Q_d)\setminus E(G)$ can be added to $G$ one at a time so that each added edge creates a new copy of $Q_m$. Answering another question of Johnson and Pinto, we determine the minimum number of edges in a weakly $(Q_d,Q_m)$-saturated graph for all $d\geq m\geq1$. More generally, we determine the minimum number of edges in a subgraph of the $d$-dimensional grid $P_k^d$ which is weakly saturated with respect to `axis aligned' copies of a smaller grid $P_r^m$. We also study weak saturation of cycles in the grid.