H-percolation with a random H

Abstract

In H-percolation, we start with an Erdős–Rényi graph ${\mathcal{G}}_{n,p}$ and then iteratively add edges that complete copies of H. The process percolates if all edges missing from ${\mathcal{G}}_{n,p}$ are eventually added. We find the critical threshold $p_c$ when $H={\mathcal{G}}_{k,1∕2}$ is uniformly random, solving a problem of Balogh, Bollobás and Morris. In this sense, we find $p_c$ for most graphs H.