Abstract. We present a complete analytic solution to the problem of counting planar graphs. We prove an estimate gn ∼ g ·n −7/2 γ n n! for the number gn 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) · n −4 γ(q) n n! for the number of planar graphs with n vertices and ⌊qn⌋ 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 (ν), where ν 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.

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We consider graph classes G in which every graph has components in a class C of connected graphs. We provide a framework for the asymptotic study of |G n,N |, the number of graphs in G with n vertices and N := λn components, where λ ∈ (0, 1). Assuming that the number of graphs with n vertices in C satisfies |C n | ∼ bn −(1+α) ρ −n n!, n → ∞ for some b, ρ > 0 and α > 1-a property commonly encountered in graph enumeration-we show that |G n,N | ∼ c(λ)n f (λ) (log n) g(λ) ρ −n h(λ) N n! N ! , n → ∞ for explicitly given c(λ), f (λ), g(λ) and h(λ). These functions are piecewise continuous with a discontinuity at a critical value λ * , which we also determine. The central idea in our approach is to sample objects of G randomly by so-called Boltzmann generators in order to translate enumer-ative problems to the analysis of iid random variables. By that we are able to exploit local limit theorems and large deviation results well-known from probability theory to prove our claims. The main results are formulated for generic com-binatorial classes satisfying the SET-construction.

International audienceIt is shown that the number of labelled graphs with $n$ vertices that can be embedded in the orientable surface $\mathbb{S}_g$ of genus $g$ grows asymptotically like $ c^{(g)}n^{5(g-1)/2-1}\gamma^n n! $, where $c^{(g)} >0$, and $\gamma \approx 27.23$ is the exponential growth rate of planar graphs. This generalizes the result for the planar case $g=0$, obtained by Giménez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in $\mathbb{S}_g$ has a unique 2-connected component of linear size with high probability

Abstract. We show that the number of vertices of a given degree k in several kinds of series-parallel labelled graphs of size n satisfy a central limit theorem with mean and variance proportional to n, and quadratic exponential tail estimates. We further prove a corresponding theorem for the number of nodes of degree two in labelled planar graphs. The proof method is based on generating functions and singularity analysis. In particular we need systems of equations for multivariate generating functions and transfer results for singular representations of analytic functions. Statement of main resultsA graph is series-parallel if it does not contain the complete graph K 4 as a minor; equivalently, if it does not contain a subdivision of K 4 . Since both K 5 and K 3,3 contain a subdivision of K 4 , by Kuratowski's theorem a series-parallel graph is planar. An outerplanar graph is a planar graph that can be embedded in the plane so that all vertices are incident to the outer face. They are characterized as those graphs not containing a minor isomorphic to (or a subdivision of) either K 4 or K 2,3 . These are important subfamilies of planar graphs, as they are much simpler but often they already capture the essential structural properties of planar graphs. In particular, they are used as a natural first benchmark for many algorithmic problems and conjectures related to planar graphs The purpose of this paper is to study the number of vertices of given degree in certain classes of labelled planar graphs. In particular, we study labelled outerplanar graphs and series-parallel graphs; in what follows, all graphs are labelled.In order to state our results we introduce the notion of the degree distribution of a random outerplanar graph (the definition for series-parallel graphs is exactly the same). For every n we consider the class of all vertex labelled outerplanar graphs with n vertices. Let D n denote the degree of a randomly chosen vertex in this class of graphs 1 . Then we say that this class of graphs has a degree distribution if there exist non-negative numbers d k with k≥0 d k = 1 such that for all kIn a companion paper [6], we have established that the classes of 2-connected, connected or all outerplanar graphs, as well as the corresponding classes of seriesparallel graphs have a degree distribution. We describe briefly the degree distribution in the outerplanar case, which is the simplest one, and refer to [6] for the other 1 Alternatively we can define Dn as the degree of the vertex with label 1.