Abstract
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 ε > 0, if $p=\frac{1+\epsilon}{r}$, then Gp is non-planar with probability approaching 1 as r grows. This generalizes classical results on planarity of binomial random graphs.
Highlights
Planarity is a fairly classical subject in the theory of random graphs
Already Erdos and Renyi in their groundbreaking paper [2] stated that a random graph Gn,p has a sharp threshold for non-planarity at p = 1/n in the following sense: if p = c/n and c < 1 the random graph Gn,p is with high probability planar, while for c > 1 Gn,p is whp non-planar
The ErdosRenyi argument for non-planarity had a certain inaccuracy, as was pointed by Luczak and Wierman [9], who explained how the probable non-planarity result can be obtained by other means
Summary
Planarity is a fairly classical subject in the theory of random graphs. Already Erdos and Renyi in their groundbreaking paper [2] stated (re-casting their statement in the language of binomial random graphs) that a random graph Gn,p has a sharp threshold for non-planarity at p = 1/n in the following sense: if p = c/n and c < 1 the random graph Gn,p is with high probability (whp) planar, while for c > 1 Gn,p is whp non-planar. The aim of this paper is to generalize this classical non-planarity result to a much wider class of probability spaces. For a graph G = (V, E) and 0 ≤ p ≤ 1 we can define the random graph Gp = (V, Ep) where each e ∈ E. is independently included in Ep with probability p.
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