Ore and Chvátal‐type degree conditions for bootstrap percolation from small sets

Abstract

AbstractBootstrap percolation is a deterministic cellular automaton in which vertices of a graph begin in one of two states, “dormant” or “active.” Given a fixed positive integer , a dormant vertex becomes active if at any stage it has at least active neighbors, and it remains active for the duration of the process. Given an initial set of active vertices , we say that G ‐percolates (from ) if every vertex in becomes active after some number of steps. Let denote the minimum size of a set such that G ‐percolates from . Bootstrap percolation has been studied in a number of settings and has applications to both statistical physics and discrete epidemiology. Here, we are concerned with degree‐based density conditions that ensure . In particular, we give an Ore‐type degree sum result that states that if a graph satisfies , then either or is in one of a small number of classes of exceptional graphs. (Here, is the minimum sum of degrees of two nonadjacent vertices in .) We also give a Chvátal‐type degree condition: If is a graph with degree sequence such that or for all , then or falls into one of several specific exceptional classes of graphs. Both of these results are inspired by, and extend, an Ore‐type result in a paper by Freund et al.