2022/09/15 by Dukes, Peter J., Noel, Jonathan A., Romer, Abel E.
#05C35 #05D99 #82B43 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2209.07594
For r≥1, the r-neighbour bootstrap process in a graph G starts with a set of infected vertices and, in each time step, every vertex with at least r infected neighbours becomes infected. The initial infection percolates if every vertex of G is eventually infected. We exactly determine the minimum cardinality of a set that percolates for the 3-neighbour bootstrap process when G is a 3-dimensional grid with minimum side-length at least 11. We also characterize the integers a and b for which there is a set of cardinality (ab+a+b)/(3) that percolates for the 3-neighbour bootstrap process in the a× b grid; this solves a problem raised by Benevides, Bermond, Lesfari and Nisse [HAL Research Report 03161419v4, 2021].