# Small Percolating Sets

## Document Type

Presentation Abstract

## Presentation Date

10-21-2019

## Abstract

Bootstrap percolation is a simple monotone cellular automaton which was originally introduced by Chalupa, Leath and Reich as a model of ferromagnetism in the late 1970s. In this model, we think of some vertices of a graph as being initially *infected*. Even worse, this infection can spread -- an *uninfected* vertex with many infected neighbors will itself become infected. Does the infection spread to the entire graph? Will it stop? Can it be efficiently quarantined?

In this talk, we give an introduction to bootstrap percolation and its history, highlighting a few major breakthroughs, classic problems, and important variants. Then, we'll proceed to a simple sounding extremal problem -- which graphs have a small set of vertices whose infection will eventually spread to the entire graph? This question was the topic of this summer's Graph Brain Project; we will describe several results which came out of the summer's work, as well as that workshop's somewhat unusual characteristics.

No background knowledge will be assumed -- the aim of this talk is to introduce you to the area and its problems, rather than to show complicated proofs*.*

## Recommended Citation

Bushaw, Neal, "Small Percolating Sets" (2019). *Colloquia of the Department of Mathematical Sciences*. 579.

https://scholarworks.umt.edu/mathcolloquia/579

## Additional Details

Monday, October 21, 2019 at 3:00 p.m. in Math 103

Refreshments at 4:00 p.m. in Math Lounge 109