Contagious sets in dense graphs

Abstract

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 on n vertices.We examine density conditions that ensure m(G,r)=r for all r, and first show a necessary and sufficient condition on the minimum degree. Moreover, we study the speed with which the activation spreads and provide tight upper bounds on the number of rounds it takes until all nodes are activated in such graphs.We also investigate what average degree asserts the existence of small contagious sets. 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.