Bootstrap percolation in Ore-type graphs

TL;DR

This paper establishes Ore-type degree conditions that guarantee the existence of small percolating sets in graphs for the bootstrap infection process, extending previous minimum degree results to broader scenarios.

Contribution

It provides new Ore-type conditions ensuring small percolating sets in graphs, generalizing earlier minimum degree results for bootstrap percolation.

Findings

Conditions for small percolating sets based on degree sums

Extension of Gunderson's minimum degree results

Bounds for large percolating sets in degree terms

Abstract

The $r$-neighbour bootstrap process describes an infection process on a graph, where we start with a set of initially infected vertices and an uninfected vertex becomes infected as soon as it has $r$ infected neighbours. An inital set of infected vertices is called percolating if at the end of the bootstrap process all vertices are infected. We give Ore-type conditions that guarantee the existence of a small percolating set of size $l\leq 2r-2$ if the number of vertices $n$ of our graph is sufficiently large: if $l\geq r$ and satisfies $2r \geq l+2 \lfloor \sqrt{2(l-r)+0.25}+2.5 \rfloor-1$ then there exists a percolating set of size $l$ for every graph in which any two non-adjacent vertices $x$ and $y$ satisfy $deg(x)+deg(y) \geq n+4r-2l-2\lfloor\sqrt{2(l-r)+0.25}+2.5 \rfloor-1$ and if $l$ is larger with $l\leq 2r-2$ there exists a percolating set of size $l$ if $deg(x)+deg(y) \geq n+2r-l-2$. Our results extend the work of Gunderson, who showed that a graph with minimum degree $\lfloor n/2 \rfloor+r-3$ has a percolating set of size $r \geq 4$. We also give bounds for arbitrarily large $l$ in the minimum degree setting.