Published online by Cambridge University Press: 31 March 2016
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(2d).
We also study weak saturation, which is a form of bootstrap percolation. A spanning subgraph of Qd is said to be weakly (Qd, Qm)-saturated if the edges of E(Qd)\E(G) can be added to G one at a time so that each added edge creates a new copy of Qm. 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 Pkd which is weakly saturated with respect to ‘axis aligned’ copies of a smaller grid Prm. We also study weak saturation of cycles in the grid.