# Minimum degree conditions for small percolating sets in bootstrap percolation

## TL;DR

This paper establishes precise minimum degree thresholds in large graphs that guarantee the existence of small percolating sets in bootstrap percolation, advancing understanding of infection spread conditions.

## Contribution

It provides sharp minimum degree conditions for the existence of small percolating sets in bootstrap percolation for all r ≥ 3, filling a gap in percolation theory.

## Key findings

- For r=3, minimum degree > n/2 guarantees a percolating set of size 3.
- For r ≥ 4, minimum degree > n/2 + (r-3) guarantees a percolating set of size r.
- Results are sharp, with examples demonstrating optimality.

## Abstract

The $r$-neighbour bootstrap process is an update rule for the states of vertices in which `uninfected' vertices with at least $r$ `infected' neighbours become infected and a set of initially infected vertices is said to \emph{percolate} if eventually all vertices are infected. For every $r \geq 3$, a sharp condition is given for the minimum degree of a sufficiently large graph that guarantees the existence of a percolating set of size $r$. In the case $r=3$, for $n$ large enough, any graph on $n$ vertices with minimum degree $\lfloor n/2 \rfloor +1$ has a percolating set of size $3$ and for $r \geq 4$ and $n$ large enough (in terms of $r$), every graph on $n$ vertices with minimum degree $\lfloor n/2 \rfloor + (r-3)$ has a percolating set of size $r$. A class of examples are given to show the sharpness of these results.