$K_{r,s}$ graph bootstrap percolation

TL;DR

This paper investigates the percolation process in Erdős-Rényi random graphs driven by the formation of complete bipartite subgraphs, determining thresholds for balanced cases and providing bounds for all parameters.

Contribution

It determines the percolation threshold for balanced $K_{r,s}$ in Erdős-Rényi graphs up to a logarithmic factor and establishes a general lower bound for all $K_{r,s}$.

Findings

Percolation threshold for balanced $K_{r,s}$ determined up to a logarithmic factor.

Partial answer to a question by Balogh, Bollobás, and Morris.

General lower bound for all $K_{r,s}$ with $r \\geq s \\geq 3$.

Abstract

A graph $G$ percolates in the $K_{r,s}$-bootstrap process if we can add all missing edges of $G$ in some order such that each edge creates a new copy of $K_{r,s}$, where $K_{r,s}$ is the complete bipartite graph. We study $K_{r,s}$-bootstrap percolation on the Erd\H{o}s-R\'{e}nyi random graph, and determine the percolation threshold for balanced $K_{r,s}$ up to a logarithmic factor. This partially answers a question raised by Balogh, Bollob\'as, and Morris. We also establish a general lower bound of the percolation threshold for all $K_{r,s}$, with $r\geq s \geq 3$.