Open Access
2016 Connectivity threshold for random subgraphs of the Hamming graph
Lorenzo Federico, Remco van der Hofstad, Tim Hulshof
Electron. Commun. Probab. 21: 1-8 (2016). DOI: 10.1214/16-ECP4479


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.


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Lorenzo Federico. Remco van der Hofstad. Tim Hulshof. "Connectivity threshold for random subgraphs of the Hamming graph." Electron. Commun. Probab. 21 1 - 8, 2016.


Received: 13 August 2015; Accepted: 23 February 2016; Published: 2016
First available in Project Euclid: 14 March 2016

zbMATH: 1336.05074
MathSciNet: MR3485396
Digital Object Identifier: 10.1214/16-ECP4479

Primary: 05C40 , 60K35 , 82B43

Keywords: connectivity threshold , critical window , percolation , random graph

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