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 positive 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 σ2(G)≥n−2, then either m(G,2)=2 or G is in one of a small number of classes of exceptional graphs. (Here, σ2(G) is the minimum sum of degrees of two nonadjacent vertices in G.) We also give a Chvátal‐type degree condition: If G is a graph with degree sequence d1≤d2≤⋯≤dn such that di≥i+1 or dn−i≥n−i−1 for all 1≤i<n2, 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 a paper by Freund et al.

Minimum Degree Conditions for Small Percolating Sets in Bootstrap Percolation

“…One could ask for the conditions on δ(G) that guarantee m(G, r) k for a fixed k r+1. Following the line of inquiry in [9] and [11], one might consider the lower bounds on σ 2 (G) that guarantee that m(G, r) = r for r 3. After this paper was submitted, answers to these questions for k 2r − 2 were given by Wesolek [21].…”

“…Defining σ 2 (G) to be the minimum sum of degrees of non-adjacent vertices in G, they showed that for a graph on n 2 vertices, if σ 2 (G) n, then m(G, 2) = 2. Subsequently, Dairyko, Ferrara, Lidický, Martin, Pfender, and Uzzell [9] improved this result showing that, except for a list of exceptional graphs that they completely characterized, if σ 2 (G) n − 2, then m(G, 2) = 2. Their results show that the only graph with δ(G) = |V (G)|/2 and m(G, 2) > 2 is the 5-cycle.…”

3-Neighbor bootstrap percolation on grids

“…A minimum r-percolating set in G is an r-percolating set S of G satisfying m(G, r) = |S|. Bootstrap percolation is very well studied in graphs, see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][23][24][25][26][27].…”

Contagious sets in a degree‐proportional bootstrap percolation process

“…Additionally, Morrison and Noel also proved bounds for multidimensional rectangular grids. Furthermore, the minimal size of a contagious set has been investigated for expander graphs , very dense graphs , random graphs , with additional Ore and Chvátal‐type degree conditions , and in the setting of hypergraphs .…”