Degree distribution in random planar graphs

We prove that for each k >= 0, the probability that a root vertex in a random planar graph. has degree k tends to a computable constant d(k); so that the expected number of vertices of degree k is asymptotically d(k)n, and moreover that Sigma(k)dk = 1. The proof uses the tools developed by Gimenez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise.. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function P(w) = Sigma(k)d(k)w(k). From this we can compute the dk to any degree of accuracy, and derive the asymptotic estimate dk similar to c.k(-1/2)q(k) for large values of k, where q approximate to 0.67 is a constant defined analytically. (C) 2011 Elsevier Inc. All rights reserved.