In this paper, the scaling limit of random connected cubic planar graphs (respectively
multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main …

We survey several results on the enumeration of planar graphs and on properties of random
planar graphs. This includes basic parameters, such as the number of edges and the …

Let $ F (N, m) $ denote a random forest on a set of $ N $ vertices, chosen uniformly from all
forests with $ m $ edges. Let $ F (N, p) $ denote the forest obtained by conditioning the …

We provide precise asymptotic estimates for the number of several classes of labeled cubic
planar graphs, and we analyze properties of such random graphs under the uniform …

The theory of random graphs deals with asymptotic properties of graphs equipped with a
certain probability distribution; for example, it studies how the component structure of a …

Let $ G (n, M) $ be the uniform random graph with $ n $ vertices and $ M $ edges. Erdős and
Rényi (1960) conjectured that the limiting probability\[\lim _ {n\to\infty}\mathrm {Pr}\{G …

Let be the orientable surface of genus and denote by the class of all graphs on vertex set
with edges embeddable on. We prove that the component structure of a graph chosen …

Let $\mathbb {S} _g $ be the orientable surface of genus $ g $. We show that the number of
vertex-labelled cubic multigraphs embeddable on $\mathbb {S} _g $ with $2 n $ vertices is …

The Erdős‐Rényi process begins with an empty graph on n vertices, with edges added
randomly one at a time to the graph. A classical result of Erdős and Rényi states that the …

We investigate the genus g (n, m) of the Erdős‐Rényi random graph G (n, m), providing a
thorough description of how this relates to the function m= m (n), and finding that there is …