On the Non-Planarity of a Random Subgraph

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

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