The family of well-orderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a well-orderly map. We show that the number of well-orderly maps with n nodes is at most 2 αn + O (log n ), where α≈4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log2 p(n)≤4.91n.
The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges.
Finally we obtain as an outcome of our combinatorial analysis an explicit linear-time encoding algorithm for unlabeled planar graphs using, in the worst-case, a rate of 4.91 bits per node and of 2.82 bits per edge.