Ore and Chv\'atal-type Degree Conditions for Bootstrap Percolation from Small Sets

Abstract

Bootstrap percolation is a deterministic cellular automaton in which vertices of a graph~$G$ begin in one of two states, "dormant" or "active". Given a fixed integer $r$, a dormant vertex becomes active if at any stage it has at least $r$ active neighbors, and it remains active for the duration of the process. Given an initial set of active vertices $A$, we say that $G$ $r$-percolates (from $A$) if every vertex in $G$ becomes active after some number of steps. Let $m(G,r)$ denote the minimum size of a set $A$ such that $G$ $r$-percolates from $A$. 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 $m(G,2)=2$. In particular, we give an Ore-type degree sum result that states that if a graph $G$ satisfies $\sigma_2(G)\ge n-2$, then either $m(G,2)=2$ or $G$ is in one of a small number of classes of exceptional graphs. We also give a Chv\'{a}tal-type degree condition: If $G$ is a graph with degree sequence $d_1\le d_2\le\dots\le d_n$ such that $d_i \geq i+1$ or $d_{n-i} \geq n-i-1$ for all $1 \leq i < \frac{n}{2}$, then $m(G,2)=2$ or $G$ falls into one of several specific exceptional classes of graphs. Both of these results are inspired by, and extend, an Ore-type result in [D. Freund, M. Poloczek, and D. Reichman, Contagious sets in dense graphs, to appear in European J. Combin.]