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.
Highlights
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 find that the threshold does not depend on d, unlike the phase transition of the giant connected component of the Hamming graph
In this paper we investigate the random edge subgraph of d−dimensional Hamming graphs
Summary
In this paper we investigate the random edge subgraph of d−dimensional Hamming graphs. Hamming graphs are defined as follows: Definition 1.1 (Hamming graph). We define the d−dimensional Hamming graph H(d, n) as the graph with vertex set. We study a percolation model on the Hamming graph. The analogous problem was first solved for the Erdos-Rényi Random Graph (ERRG) in [3]. [5, Section 5.3]), but we find that at places the internal geometry of the Hamming graph plays an important role. To overcome this difficulty we use an induction on the dimension d and an exploration of the graph
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