Contagious Sets in Dense Graphs

We study the activation process in undirected graphs known as bootstrap percolation: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it had at least r active neighbors, for a threshold r that is identical for all vertices. A contagious set is a vertex set whose activation results with the entire graph being active. Let m(G, r) be the size of a smallest contagious set in a graph G. We examine density conditions that ensure that a given n-vertex graph G = (V,E) has a small contagious set. With respect to the minimum degree, we prove that if G has minimum degree n/2 then m(G, 2) = 2. We also provide tight upper bounds on the number of rounds until all nodes are active. For n ≥ k ≥ r, we denote by M(n, k, r) the maximum number of edges in an n-vertex graph G satisfying m(G, r) > k. We determine the precise value of M(n, k, 2) and M(n, k, k) assuming that n is sufficiently large compared to k.