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_pindependently and with probability p. We prove that for any constant ε > 0, if , then G_p is non-planar with probability approaching 1 as r grows. This generalizes classical results on planarity of binomial random graphs.