On the non-planarity of a random subgraph

Let $G$ be a finite graph with minimum degree $r$. Form a random subgraph $G_p$ of $G$ by taking each edge of $G$ into $G_p$ independently and with probability $p$. We prove that for any constant $ε>0$, if $p=\frac{1+ε}{r}$, then $G_p$ is non-planar with probability approaching 1 as $r$ grows. This generalizes classical results on planarity of binomial random graphs.

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