On $K_{2,t}$-bootstrap percolation

TL;DR

This paper investigates the percolation thresholds in the $K_{2,t}$-bootstrap process, providing bounds and a specific threshold function for the case when $t=4$, advancing understanding of bootstrap percolation in bipartite graphs.

Contribution

It establishes new lower and upper bounds on the percolation threshold for $K_{2,t}$-bootstrap percolation and derives an explicit threshold function for $K_{2,4}$.

Findings

Derived bounds on the percolation threshold for $K_{2,t}$.

Established a threshold function for $K_{2,4}$-bootstrap percolation.

Enhanced understanding of bootstrap percolation in bipartite graphs.

Abstract

Given two graphs $G$ and $H$, it is said that $G$ percolates in $H$-bootstrap process if one could join all the nonadjacent pairs of vertices of $G$ in some order such that a new copy of $H$ is created at each step. Balogh, Bollob\'as and Morris in 2012 investigated the threshold of $H$-bootstrap percolation in the Erd\H{o}s-R\'enyi model for the complete graph $H$ and proposed the similar problem for $H=K_{s,t}$, the complete bipartite graph. In this paper, we provide lower and upper bounds on the threshold of $K_{2, t}$-bootstrap percolation. In addition, a threshold function is derived for $K_{2, 4}$-bootstrap percolation.