Planar Graphs, via Well-Orderly Maps and Trees
Résumé
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 $2^{\alpha n + O(\log n)}$, where $\alpha \approx 4.91$. A direct consequence of this is a new upper bound on the number $p(n)$ of unlabeled planar graphs with $n$ nodes, $\log_2 p(n) \leq 4.91n$. 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.