Bootstrap percolation of extension hypergraphs

TL;DR

This paper investigates the maximum running time of a bootstrap percolation process on hypergraphs, establishing asymptotic bounds for a broad class of such structures based on graph extensions.

Contribution

It determines the asymptotics of the maximum running time for a large class of hypergraphs derived from graph extensions, resolving a problem posed by Bollobás.

Findings

Maximum running time is bounded by a constant depending only on the graph size and hypergraph uniformity.

Asymptotic bounds are established for the $F^{(k)}(G)$ hypergraphs.

The study extends previous work by Noel and Ranganathan on hypergraph bootstrap percolation.

Abstract

For $k$-graphs $F$ and $H_0$ the $F$-bootstrap percolation process (or $F$-process) starting with $H_0$ is a sequence $(H_i)_{i\geq0}$ of $k$-graphs such that $H_{i+1}$ is obtained from $H_i$ by adding all those $e\in V(H_0)^{(k)}\setminus E(H_i)$ as edges that complete a new copy of $F$. The running time of this $F$-process, denoted by $M_F(H_0)$, is the smallest $i$ with $H_i=H_{i+1}$. Bollob\'as proposed the problem of determining the maximum running time for $n\in\mathbb{N}$, i.e., $$M_F(n)=\max_{\vert V(H_0)\vert=n}M_F(H_0)\,.$$ Recently, Noel and Ranganathan initiated the study of this quantity for $k$-graphs. In this work, we determine the asymptotics of $M_F(n)$ for a large class of $k$-graphs. Given a graph $G=(V,E)$, the $k$-extension of $G$ is a $k$-graph $F^{(k)}(G)$ obtained from $G$ by enlarging each edge with a $(k-2)$-set of new vertices. We show that for every graph $G$ on $t$ vertices and every $k\geq 3$, $M_{F^{(k)}(G)}(n)\leq C_{k,t}$ for some constant $C_{k,t}$ depending only on $t$ and $k$.