Asymptotic enumeration and limit laws of planar graphs
Abstract
We present a complete analytic solution to the problem of counting planar graphs. We prove an estimate g_n ̃ g\cdot n^{7/2} γ^n n! for the number g_n of labelled planar graphs on n vertices, where γ and g are explicit computable constants. We show that the number of edges in random planar graphs is asymptotically normal with linear mean and variance and, as a consequence, the number of edges is sharply concentrated around its expected value. Moreover we prove an estimate g(q)\cdot n^{4}γ(q)^n n! for the number of planar graphs with n vertices and lfloor qn rfloor edges, where γ(q) is an analytic function of q . We also show that the number of connected components in a random planar graph is distributed asymptotically as a shifted Poisson law 1+P(nu) , where nu is an explicit constant. Additional Gaussian and Poisson limit laws for random planar graphs are derived. The proofs are based on singularity analysis of generating functions and on perturbation of singularities.
 Publication:

Journal of the American Mathematical Society
 Pub Date:
 April 2009
 DOI:
 10.1090/S0894034708006243
 Bibcode:
 2009JAMS...22..309G
 Keywords:

 Planar graph;
 random planar graph;
 asymptotic enumeration;
 limit law;
 normal law;
 analytic combinatorics.