Ore and Chvátal-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 Chvátal-type degree condition: If $G$ is a graph with degree sequence $d_1\le d_2\le \cdots \le d_n$ such that $d_i\ge i+1$ or $d_{n-i}\ge n-i-1$ for all $1\le i<\frac{n}{2}$, 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.]

15 pages, 3 figures

References in corpus (3)

• Resilience to Contagion in Financial Networks
• Bootstrap percolation on the random graph $G_{n,p}$
• Irreversible 2-conversion set in graphs of bounded degree

Cited by in corpus (1)

• Bootstrap percolation in Ore-type graphs