Geodesic
Random subgraphs in Cartesian powers of regular graphs
The electronic journal of combinatorics, Tome 19 (2012) no. 1
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Let $G$ be a connected $d$-regular graph with $k$ vertices. We investigate the behaviour of a spanning random subgraph $G^n_p$ of $G^n$, the $n$-th Cartesian power of $G$, which is constructed by deleting each edge independently with probability $1-p$. We prove that $\lim\limits_{n \rightarrow \infty} \mathbb{P}[G^n_p {\rm \ is \ connected}]=e^{-\lambda}$, if $p=p(n)=1-\left(\frac{\lambda_n^{1/n}}{k}\right)^{1/d}$ and $\lambda_n \rightarrow \lambda>0$ as $n \rightarrow \infty$. This extends a result of L. Clark, Random subgraphs of certain graph powers, Int. J. Math. Math. Sci., 32(5):285-292, 2002.
DOI : 10.37236/2058
Classification : 05C80, 05C76 
@article{10_37236_2058,
     author = {Felix Joos},
     title = {Random subgraphs in {Cartesian} powers of regular graphs},
     journal = {The electronic journal of combinatorics},
     year = {2012},
     volume = {19},
     number = {1},
     doi = {10.37236/2058},
     zbl = {1243.05220},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/2058/}
}
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Felix Joos. Random subgraphs in Cartesian powers of regular graphs. The electronic journal of combinatorics, Tome 19 (2012) no. 1. doi: 10.37236/2058

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