## Asymptotic enumeration and limit laws of planar graphs

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- by Omer Giménez and Marc Noy
- J. Amer. Math. Soc.
**22**(2009), 309-329 - DOI: https://doi.org/10.1090/S0894-0347-08-00624-3
- Published electronically: October 17, 2008
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## Abstract:

We present a complete analytic solution to the problem of counting planar graphs. We prove an estimate $g_n \sim g\cdot n^{-7/2} \gamma ^n n!$ for the number $g_n$ of labelled planar graphs on $n$ vertices, where $\gamma$ 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 $g(q)\cdot n^{-4}\gamma (q)^n n!$ for the number of planar graphs with $n$ vertices and $\lfloor qn \rfloor$ edges, where $\gamma (q)$ 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.## References

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*Analytic combinatorics*, to be published in 2008 by Cambridge University Press, preliminary version available at http://algo.inria.fr/flajolet/Publications.

## Bibliographic Information

**Omer Giménez**- Affiliation: Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya, Jordi Girona 1–3, 08034 Barcelona, Spain
- Email: Omer.Gimenez@upc.edu
**Marc Noy**- Affiliation: Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Jordi Girona 1–3, 08034 Barcelona, Spain
- Email: Marc.Noy@upc.edu
- Received by editor(s): August 3, 2005
- Published electronically: October 17, 2008
- Additional Notes: The first author’s research was supported in part by Project MTM2005-08618C02-01.
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**22**(2009), 309-329 - MSC (2000): Primary 05A16, 05C30; Secondary 05C10
- DOI: https://doi.org/10.1090/S0894-0347-08-00624-3
- MathSciNet review: 2476775