2026/08/06 by Neal Bushaw, Alexander Clifton
Mathematics · #math.CO #math.PR #msc:05C35 #msc:05D99
39 pages, 23 figures.x
arxiv created 2026/08/06 · arxiv updated 2026/08/07
In the 3-neighbor bootstrap percolation process, a vertex becomes (and remains) infected if at least three of its neighbors are infected. We say that an initial configuration of infected vertices percolates if eventually all vertices are infected. We exactly determine the size of the minimum percolating set for the 3-neighbor bootstrap percolation process on all remaining open cases for rectangular grid graphs Pm\square Pn. This extends earlier work of Dukes, Noel, and Romer. Additionally, we consider the same question for the toroidal grids Cm\square Cn, proving upper and lower bounds which are at most one apart and determining the answer precisely in many divisibility cases.