Saturation in the Hypercube and Bootstrap Percolation

Let Qd denote the hypercube of dimension d. Given d ≥ m, a spanning subgraph G of Qd is said to be (Qd, Qm)-saturated if it does not contain Qm as a subgraph but adding any edge of E(Qd) \E(G) creates a copy of Qm in G. Answering a question of Johnson and Pinto [27], we show that for every fixed m ≥ 2 the minimum number of edges in a (Qd, Qm)-saturated graph is Θ(2 d). We also study weak saturation, which is a form of bootstrap percolation. Given graphs F and H, a spanning subgraph G of F is said to be weakly (F,H)-saturated if the edges of E(F )\E(G) can be added to G one at a time so that each additional edge creates a new copy of H. Answering another question of Johnson and Pinto [27], we determine the minimum number of edges in a weakly (Qd, Qm)-saturated graph for all d ≥ m ≥ 1. More generally, we determine the minimum number of edges in a subgraph of the d-dimensional grid P d k which is weakly saturated with respect to ‘axis aligned’ copies of a smaller grid Pm r . We also study weak saturation of cycles in the grid.