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Article Dans Une Revue Graphs and Combinatorics Année : 2006

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.
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Dates et versions

hal-00159296 , version 1 (02-07-2007)

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  • HAL Id : hal-00159296 , version 1

Citer

Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse, Dominique Poulalhon, Gilles Schaeffer. Planar graphs, via well-orderly maps and trees. Graphs and Combinatorics, 2006, 22 (2), pp.185-202. ⟨hal-00159296⟩
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