Planar Graphs, via Well-Orderly Maps and Trees

Abstract

Abstract. 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, log 2 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.

Asymptotic enumeration and limit laws of planar graphs

“…The best values achieved so far were α ≈ 1.85 (shown in [8], improving upon [4]) and β ≈ 2.44 (shown in [3], improving upon [14]). Theorem 2 shows that in fact there is only one constant that matters, namely κ.…”

“…The best upper bound obtained so far is γ u < 30.06. This is proved in [3] by showing that an unlabelled planar graph with n vertices can be encoded with 4.91n bits. On the other hand, our determination of γ provides a lower bound on γ u , and shows that at least 4.76 ≈ log 2 γ bits per vertex are needed.…”

“…In [3] it is proved that an unlabelled planar graph with m edges can be encoded with 2.82m bits. We can show that at least 2.59 bits per edge are needed, as follows.…”

Convex Drawings of 3-Connected Plane Graphs

“…This bound depends on the parameter ∆ S Min which can be equal to zero in 'bad' cases. Bonichon et al [2] have analyzed the asymptotic average value of ∆ S Min over triangulations with n vertices. They prove that E(∆ S Min ) = n/8 + o(n).…”

Transversal Structures on Triangulations, with Application to Straight-Line Drawing

“…Performing simulations on objects of large size (n ≈ 50000), it was observed that the size of the grid is always approximately α) with CompactTransversalDraw, where α ≈ 0.18. It turns out that the size of the grid can be readily analyzed thanks to our closure-bijection, in the same way that bijection of [12] allowed to analyze parameters of Schnyder woods in [3]. Indeed, unused abscissas and ordinates of TransversalDraw correspond to certain inner edges of the ternary tree, whose number can be proven to be asymptotically almost surely 5n 27 up to fluctuations of order √ n. A second application is counting rooted 4-connected triangulations with n vertices, whose set is denoted by C n .…”