Using polynomials to find lower bounds for $r$-bond bootstrap percolation

TL;DR

This paper develops algebraic polynomial methods to establish lower bounds on the size of initial infected edge sets needed for complete percolation in $r$-bond bootstrap percolation across various graph families.

Contribution

It introduces recursive formulas and extends algebraic techniques to determine minimal percolating sets in multiple graph classes.

Findings

Derived recursive formulas for minimal percolating sets

Extended algebraic methods to broader graph families

Provided bounds that improve understanding of percolation thresholds

Abstract

The $r$-bond bootstrap percolation process on a graph $G$ begins with a set $S$ of infected edges of $G$ (all other edges are healthy). At each step, a healthy edge becomes infected if at least one of its endpoints is incident with at least $r$ infected edges (and it remains infected). If $S$ eventually infects all of $E(G)$, we say $S$ percolates. In this paper we provide recursive formulae for the minimum size of percolating sets in several large families of graphs. We utilise an algebraic method introduced by Hambardzumyan, Hatami, and Qian, and substantially extend and generalise their work.