Random planar graphs

Authors:
Colin McDiarmid;Angelika Steger;Dominic J. A. Welsh
Affiliations:
Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK;Institut für Theoretische Informatik, ETH Zürich, 8092 Zürich, Switzerland;Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK
Venue:
Journal of Combinatorial Theory Series B
Year:
2005

Citing 6
Cited 30

The four-colour theorem

Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
Random maps, coalescing saddles, singularity analysis, and airy phenomena

Random Structures & Algorithms - Special issue on analysis of algorithms dedicated to Don Knuth on the occasion of his (100)8th birthday
An Information-Theoretic Upper Bound of Planar Graphs Using Triangulation

STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
On random planar graphs, the number of planar graphs and their triangulations

Journal of Combinatorial Theory Series B
On the Number of Edges in Random Planar Graphs

Combinatorics, Probability and Computing
Generating labeled planar graphs uniformly at random

ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming

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
Random triangulations of planar point sets

Proceedings of the twenty-second annual symposium on Computational geometry
How randomized search heuristics find maximum cliques in planar graphs

Proceedings of the 8th annual conference on Genetic and evolutionary computation
Proper minor-closed families are small

Journal of Combinatorial Theory Series B
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
Connectivity of addable graph classes

Journal of Combinatorial Theory Series B
Dissections, orientations, and trees with applications to optimal mesh encoding and random sampling

ACM Transactions on Algorithms (TALG)
Random graphs on surfaces

Journal of Combinatorial Theory Series B
On the maximum degree of a random planar graph

Combinatorics, Probability and Computing
Colouring Random Empire Trees

MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
On the Degree Sequences of Random Outerplanar and Series-Parallel Graphs

APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Maximal biconnected subgraphs of random planar graphs

SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Random graphs from a minor-closed class

Combinatorics, Probability and Computing
The degree sequence of random graphs from subcritical classes†

Combinatorics, Probability and Computing
Uniform random sampling of planar graphs in linear time

Random Structures & Algorithms
Maximal biconnected subgraphs of random planar graphs

ACM Transactions on Algorithms (TALG)
Small graph classes and bounded expansion

Journal of Combinatorial Theory Series B
Growth constants of minor-closed classes of graphs

Journal of Combinatorial Theory Series B
On degrees in random triangulations of point sets

Proceedings of the twenty-sixth annual symposium on Computational geometry
A separator theorem for string graphs and its applications

Combinatorics, Probability and Computing
3-connected cores in random planar graphs

Combinatorics, Probability and Computing
Degree distribution in random planar graphs

Journal of Combinatorial Theory Series A
On degrees in random triangulations of point sets

Journal of Combinatorial Theory Series A
The maximum degree of random planar graphs

Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Sampling unlabeled biconnected planar graphs

ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
On the degree distribution of random planar graphs

Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Random graphs with few disjoint cycles

Combinatorics, Probability and Computing
Planar graphs, via well-orderly maps and trees

WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science

Quantified Score

Hi-index	0.00

Visualization

Abstract

We study various properties of the random planar graph Rn, drawn uniformly at random from the class Pn of all simple planar graphs on n labelled vertices. In particular, we show that the probability that Rn is connected is bounded away from 0 and from 1. We also show for example that each positive integer k, with high probability Rn has linearly many vertices of a given degree, in each embedding Rn has linearly many faces of a given size, and Rn has exponentially many automorphisms.