The giant component after percolation of product graphs

TL;DR

This paper establishes a sharp threshold for the emergence of a giant component in percolated product graphs, extending classical results to broader classes of graphs with specific degree and isoperimetric properties.

Contribution

It generalizes the 1982 hypercube percolation result to Cartesian product graphs under degree and isoperimetric constraints.

Findings

Identifies a sharp percolation threshold for giant component emergence

Extends classical hypercube percolation results to product graphs

Provides conditions based on maximum degrees and isoperimetric constants

Abstract

In this paper we show the existence of a sharp threshold for the appearance of a giant component after percolation of Cartesian products of graphs under assumptions on their maximum degrees and their isoperimetric constants. In particular, this generalises a work of Ajtai, Koml\'os and Szemer\'edi from 1982 concerning percolation of the hypercube in high dimension.