Random planar graphs

We study various properties of the random planar graph R-n drawn uniformly at random from the class P-n of all simple planar graphs on n labelled vertices. In particular, we show that the probability that R-n is connected is bounded away from 0 and from 1. We also show for example that each positive integer k, with high probability R-n has linearly many vertices of a given degree, in each embedding R-n has linearly many faces of a given size, and R-n has exponentially many automorphisms. (C) 2004 Elsevier Inc. All rights reserved.