Saturation in the Hypercube and Bootstrap Percolation

Abstract

LetQddenote the hypercube of dimensiond. Givend⩾m, a spanning subgraphGofQdis said to be (Qd,Qm)-saturated if it does not containQmas a subgraph but adding any edge ofE(Qd)\E(G) creates a copy ofQminG. Answering a question of Johnson and Pinto [27], we show that for every fixedm⩾ 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 ofQdis said to beweakly(Qd,Qm)-saturated if the edges ofE(Qd)\E(G) can be added toGone at a time so that each added edge creates a new copy ofQm. Answering another question of Johnson and Pinto [27], we determine the minimum number of edges in a weakly (Qd,Qm)-saturated graph for alld⩾m⩾ 1. More generally, we determine the minimum number of edges in a subgraph of thed-dimensional gridPkdwhich is weakly saturated with respect to ‘axis aligned’ copies of a smaller gridPrm. We also study weak saturation of cycles in the grid.