On the degree distribution of random planar graphs

Abstract

Let Pn be the class of all planar graphs with n labeled vertices, and let Pn be a graph drawn uniformly at random from Pn. In this paper we study the degree sequence of Pn. We show that with probability 1 -- o(1) the number of vertices of degree k in Pn is very close to a quantity μkn that we determine explicitly, for all k ≤ c log n and an appropriate c > 0. A similar statement is true for random biconnected planar graphs as well.The main tool in our analysis is a framework that allows us under certain conditions to derive universal results about the degree distribution of random graphs from general classes with structural constraints. In particular, we address so-called critical graph classes, which due to their intricate structure have posed significant technical difficulties in the past.