On the maximum running time in graph bootstrap percolation

TL;DR

This paper investigates the maximum duration of the graph bootstrap percolation process before stabilization, revealing different behaviors for small and large cliques, including a new maximum time for $K_4$ and super-polynomial bounds for larger cliques.

Contribution

The paper determines the maximum running time of the bootstrap percolation process for various clique sizes, highlighting new phenomena and bounds for $K_4$ and larger cliques.

Findings

Maximum time for $K_3$ is $oxed{ ext{log}_2(n-1)}$ steps.

Maximum time for $K_4$ is exactly $n-3$ steps.

For $K_r$ with $r o ext{large}$, the process can last at least $n^{2- ext{small } ext{epsilon}_r}$ steps.

Abstract

Graph bootstrap percolation is a simple cellular automaton introduced by Bollob\'as in 1968. Given a graph $H$ and a set $G \subseteq E(K_n)$ we initially "infect" all edges in $G$ and then, in consecutive steps, we infect every $e \in K_n$ that completes a new infected copy of $H$ in $K_n$. We say that $G$ percolates if eventually every edge in $K_n$ is infected. The extremal question about the size of the smallest percolating sets when $H = K_r$ was answered independently by Alon, Kalai and Frankl. Here we consider a different question raised more recently by Bollob\'as: what is the maximum time the process can run before it stabilizes? It is an easy observation that for $r=3$ this maximum is $\lceil \log_2 (n-1) \rceil $. However, a new phenomenon occurs for $r=4$ when, as we show, the maximum time of the process is $n-3$. For $r \geq 5$ the behaviour of the dynamics is even more complex, which we demonstrate by showing that the $K_r$-bootstrap process can run for at least $n^{2-\varepsilon_r}$ time steps for some $\varepsilon_r$ that tends to $0$ as $r \to \infty$.