Lower bounds for graph bootstrap percolation via properties of polynomials

TL;DR

This paper presents a new method for establishing lower bounds on the minimal size of percolating sets in graph bootstrap processes, with applications to multidimensional tori and grids, including hypercubes.

Contribution

It introduces a straightforward technique for lower bounds and applies it to solve open questions and simplify existing results in graph bootstrap percolation.

Findings

Determined the sizes of smallest percolating sets in multidimensional tori.

Provided an alternative proof for a key result on hypercubes.

Answered an open question by Morrison and Noel.

Abstract

We introduce a simple method for proving lower bounds for the size of the smallest percolating set in a certain graph bootstrap process. We apply this method to determine the sizes of the smallest percolating sets in multidimensional tori and multidimensional grids (in particular hypercubes). The former answers a question of Morrison and Noel, and the latter provides an alternative and simpler proof for one of their main results.