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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.

We present a practical algorithm for generating random regular graphs. For all d growing as a small power of n, the d-regular graphs on n vertices are generated approximately uniformly at random, in the sense that all d-regular graphs on n vertices have in the limit the same probability as n → ∞. The expected runtime for these ds is [Oscr ](nd2).

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