Random planar graphs with n nodes and a fixed number of edges

Authors:
Stefanie Gerke;Colin McDiarmid;Angelika Steger;Andreas Weißl
Affiliations:
ETH Zürich, Institute of Theoretical Computer Science, ETH Zentrum, Switzerland;Oxford University, GB, England;ETH Zürich, Institute of Theoretical Computer Science, ETH Zentrum, Switzerland;ETH Zürich, Institute of Theoretical Computer Science, ETH Zentrum, Switzerland
Venue:
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2005

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Abstract

Let P(n, m) be the class of simple labelled planar graphs with n nodes and m edges, and let Rn,q be a graph drawn uniformly at random from P(n, [qn]). We show properties that hold with high probability (w.h.p.) for Rn,q when 1 q Rn,q contains w.h.p. linearly many nodes of each given degree and linearly many node disjoint copies of each given fixed connected planar graph. Additionally, we show that the probability that Rn,q is connected is bounded away from one by a non-zero constant. As a tool we show that (|P(n, [qn])|/n!)1/n tends to a limit as n tends to infinity.