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 ⋅ n − 7 / 2 γ n n ! g_n \sim g\cdot n^{-7/2} \gamma ^n n! for the number g n g_n of labelled planar graphs on n n vertices, where γ \gamma and g 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 ) ⋅ n − 4 γ ( q ) n n ! g(q)\cdot n^{-4}\gamma (q)^n n! for the number of planar graphs with n n vertices and ⌊ q n ⌋ \lfloor qn \rfloor edges, where γ ( q ) \gamma (q) is an analytic function of q 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 ( ν ) 1+P(\nu ) , where ν \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.