On the Number of Edges in Random Planar Graphs

Authors:
Stefanie Gerke;Colin McDiarmid
Affiliations:
TU München, Institut für Informatik, Arcisstraße 21, 80290 München, Germany (e-mail: gerke@in.tum.de);University of Oxford, Department of Statistics, South Parks Road, Oxford OX1 3TG, UK (e-mail: cmcd@stats.ox.ac.uk)
Venue:
Combinatorics, Probability and Computing
Year:
2004

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Abstract

We consider random planar graphs on $n$ labelled nodes, and show in particular that if the graph is picked uniformly at random then the expected number of edges is at least $\frac{13}{7}n +o(n)$. To prove this result we give a lower bound on the size of the set of edges that can be added to a planar graph on $n$ nodes and $m$ edges while keeping it planar, and in particular we see that if $m$ is at most $\frac{13}{7}n - c$ (for a suitable constant~$c$) then at least this number of edges can be added.