Lower bounds for graph bootstrap percolation via properties of polynomials

Extremal Bounds for Three-Neighbour Bootstrap Percolation in Dimensions Two and Three

“…This bound is not new. Variants of it appear in [9,12,29,30], for example, and several generalizations are proved in [22,28]. We provide a proof here for the sake of completeness.…”

“…. , a d ); r) can be derived from the combination of [28, Equation (1.5)] and [22,Theorem 8]. In the case d = r = 3, this bound reduces to…”

“…A natural extremal problem is to determine the cardinality of the smallest set of vertices which percolates under the r-neighbour process in a graph G, which is denoted by m(G; r). Due to its origins in statistical physics, a central focus in the study of bootstrap percolation has been on the case that G is a multidimensional grid [2][3][4][5][6][7][8][9][10][11][13][14][15][16]18,19,[21][22][23][24][25][28][29][30][31]33,34]. A beautiful exercise (featured in, e.g., [12,Problem 34]) is to show that m(n, n; 2) = n (1.1) for all n ≥ 1.…”

“…For most choices of parameters, the best known lower bound is given by [28, Theorem 4.2 and (1.5)] (see [22] for an alternative proof and several extensions). In [28], it was proved that this lower bound is tight in the case that r = 3, d ≥ 3 and…”

Deterministic bootstrap percolation on trees

“…We should also mention that in addition to the standard bootstrap percolation considered in this paper, there are also several variants. For example, two-way bootstrap percolation, which has been considered in [21,27,28], and the previously mentioned bootstrap percolation on edges, which has been recently considered in [14].…”