Connectivity threshold for random subgraphs of the Hamming graph

Abstract

We study the connectivity of random subgraphs of the $d$-dimensional Hamming graph $H(d, n)$, which is the Cartesian product of $d$ complete graphs on $n$ vertices. We sample the random subgraph with an i.i.d. Bernoulli bond percolation on $H(d,n)$ with parameter $p$. We identify the window of the transition: when $ np- \log n \to - \infty $ the probability that the graph is connected tends to $0$, while when $ np- \log n \to + \infty $ it converges to $1$. We also investigate the connectivity probability inside the critical window, namely when $ np- \log n \to t \in \mathbb{R} $. We find that the threshold does not depend on $d$, unlike the phase transition of the giant connected component of the Hamming graph (see [1]). Within the critical window, the connectivity probability does depend on $d$. We determine how.