The sharp threshold for jigsaw percolation in random graphs

Abstract

AbstractWe analyse the jigsaw percolation process, which may be seen as a measure of whether two graphs on the same vertex set are ‘jointly connected’. Bollobás, Riordan, Slivken, and Smith (2017) proved that, when the two graphs are independent binomial random graphs, whether the jigsaw process percolates undergoes a phase transition when the product of the two probabilities is$\Theta({1}/{(n\ln n)})$. We show that this threshold is sharp, and that it lies at${1}/{(4n\ln n)}$.