Asymptotic Enumeration and Limit Laws of Planar Graphs

A graph is planar if it can be embedded in the plane, or in the sphere, so that no two edges cross at an interior point. A planar graph together with a particular embedding is called a map. There is a rich theory of counting maps, started by Tutte in the 1960’s. However, in this paper we are interested in counting graphs as combinatorial objects, regardless of how many nonequivalent topological embeddings they may have. As we are going to see, this makes the counting considerably more difficult. In this paper we obtain a precise asymptotic estimate for the number of labelled planar graphs on n vertices, and we establish limit laws for several parameters in random labelled planar graphs. In particular, we show that the number of edges in random planar graphs is asymptotically normal and that the number of connected components in a random planar graph is distributed asymptotically as a shifted Poisson law. Additional Gaussian and Poisson limit laws for random planar graphs are derived. From now on, all graphs are labelled, finite and simple. Let gn be the number of planar graphs on n vertices. A superadditivity argument [12] shows that the following limit exists: γ = lim n→∞ (gn/n!) 1/n .