Discrete Mathematics & Theoretical Computer Science 
Erdős and Rényi conjectured in 1960 that the limiting probability $p$ that a random graph with $n$ vertices and $M=n/2$ edges is planar exists. It has been shown that indeed p exists and is a constant strictly between 0 and 1. In this paper we answer completely this long standing question by finding an exact expression for this probability, whose approximate value turns out to be $p ≈0.99780$. More generally, we compute the probability of planarity at the critical window of width $n^{2/3}$ around the critical point $M=n/2$. We extend these results to some classes of graphs closed under taking minors. As an example, we show that the probability of being seriesparallel converges to 0.98003. Our proofs rely on exploiting the structure of random graphs in the critical window, obtained previously by Janson, Łuczak and Wierman, by means of generating functions and analytic methods. This is a striking example of how analytic combinatorics can be applied to classical problems on random graphs.
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IsRelatedTo ARXIV 2008.09405 Source : ScholeXplorer IsRelatedTo DOI 10.1007/9783030687663_15 Source : ScholeXplorer IsRelatedTo DOI 10.48550/arxiv.2008.09405 Source : ScholeXplorer IsRelatedTo HANDLE 11590/385599
