# Percolating sets in bootstrap percolation on the Hamming graphs

## TL;DR

This paper investigates the minimal initial activation sets needed for complete activation in bootstrap percolation on Hamming graphs, providing bounds and asymptotic formulas as parameters grow large.

## Contribution

It derives new upper and lower bounds on the size of percolating sets in bootstrap percolation on Hamming graphs, including an asymptotic formula for large parameters.

## Key findings

- Established bounds on $m(K_n^d, r)$ for various parameters.
- Derived an asymptotic expression $m(K_n^d, r)=\frac{1+o(1)}{(d+1)!}r^d$ under certain growth conditions.
- Analyzed the behavior of percolating sets as $r$ and $d$ tend to infinity.

## Abstract

For any integer $r\geqslant0$, the $r$-neighbor bootstrap percolation on a graph is an activation process of the vertices. The process starts with some initially activated vertices and then, in each round, any inactive vertex with at least $r$ active neighbors becomes activated. A set of initially activated vertices leading to the activation of all vertices is said to be a percolating set. Denote the minimum size of a percolating set in the $r$-neighbor bootstrap percolation process on a graph $G$ by $m(G, r)$. In this paper, we present upper and lower bounds on $m(K_n^d, r)$, where $K_n^d$ is the Cartesian product of $d$ copies of the complete graph $K_n$ which is referred as the Hamming graph. Among other results, we show that $m(K_n^d, r)=\frac{1+o(1)}{(d+1)!}r^d$ when both $r$ and $d$ go to infinity with $r<n$ and $d=o(\!\sqrt{r})$.