Multi-coloured jigsaw percolation on random graphs

TL;DR

This paper generalizes the jigsaw percolation process to multiple graphs on the same vertex set, demonstrating a phase transition in the percolation process for random graphs based on the product of edge probabilities.

Contribution

It extends previous work by analyzing the percolation process for an arbitrary number of graphs, establishing a phase transition criterion for random graph models.

Findings

Phase transition occurs depending on the product of edge probabilities.

Generalizes previous two-graph results to multiple graphs.

Provides theoretical proof for the phase transition phenomenon.

Abstract

The jigsaw percolation process, introduced by Brummitt, Chatterjee, Dey and Sivakoff, was inspired by a group of people collectively solving a puzzle. It can also be seen as a measure of whether two graphs on a common vertex set are "jointly connected". In this paper we consider the natural generalisation of this process to an arbitrary number of graphs on the same vertex set. We prove that if these graphs are random, then the jigsaw percolation process exhibits a phase transition in terms of the product of the edge probabilities. This generalises a result of Bollob\'as, Riordan, Slivken and Smith.