On the non-planarity of a random subgraph

TL;DR

This paper proves that a random subgraph of a finite graph with high minimum degree becomes non-planar with high probability when edges are included with a probability slightly above the reciprocal of the degree, generalizing classical random graph results.

Contribution

It establishes a threshold for non-planarity in random subgraphs of graphs with high minimum degree, extending classical binomial random graph planarity results.

Findings

Random subgraphs become non-planar above a certain edge probability threshold

Probability of non-planarity approaches 1 as minimum degree increases

Generalizes classical results on binomial random graphs

Abstract

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 $\epsilon>0$, if $p=\frac{1+\epsilon}{r}$, then $G_p$ is non-planar with probability approaching 1 as $r$ grows. This generalizes classical results on planarity of binomial random graphs.