On the Number of Edges in Random Planar Graphs
Combinatorics, Probability and Computing
On the Number of Edges in Random Planar Graphs
Combinatorics, Probability and Computing
Journal of Combinatorial Theory Series B
Dissections and trees, with applications to optimal mesh encoding and to random sampling
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Random planar graphs with n nodes and a fixed number of edges
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Generating Outerplanar Graphs Uniformly at Random
Combinatorics, Probability and Computing
Generating labeled planar graphs uniformly at random
Theoretical Computer Science
The random planar graph process
Random Structures & Algorithms
Generating unlabeled connected cubic planar graphs uniformly at random
Random Structures & Algorithms
Dissections, orientations, and trees with applications to optimal mesh encoding and random sampling
ACM Transactions on Algorithms (TALG)
Journal of Combinatorial Theory Series B
Optimal coding and sampling of triangulations
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Generating labeled planar graphs uniformly at random
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Fast minor testing in planar graphs
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Degree distribution in random planar graphs
Journal of Combinatorial Theory Series A
Sampling unlabeled biconnected planar graphs
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
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Let Pn be the set of labelled planar graphs with n vertices. Denise, Vasconcellos and Welsh proved that |Pn| ≤ n! (75.8)n+o(n) and Bender, Gao and Wormald proved that |Pn| ≥ n! (26.1)n+o(n). Gerke and McDiarmid proved that almost all graphs in Pn have at least 13/7n edges. In this paper, we show that |Pn| ≤ n! (37.3)n+o(n) and that almost all graphs in Pn have at most 2.56n edges. The proof relies on a result of Tutte on the number of plane triangulations, the above result of Bender, Gao and Wormald and the following result, which we also prove in this paper: every labelled planar graph G with n vertices and m edges is contained in at least ε3(3n-m)/2 labelled triangulations on n vertices, where ε is an absolute constant. In other words, the number of triangulations of a planar graph is exponential in the number of edges which are needed to triangulate it. We also show that this bound on the number of triangulations is essentially best possible.