A calculus for the random generation of labelled combinatorial structures
Theoretical Computer Science
The distribution of the maximum vertex degree in random planar maps
Journal of Combinatorial Theory Series A
Boltzmann Samplers for the Random Generation of Combinatorial Structures
Combinatorics, Probability and Computing
Journal of Combinatorial Theory Series B
On properties of random dissections and triangulations
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
On the maximum degree of a random planar graph
Combinatorics, Probability and Computing
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
Analytic Combinatorics
Random graphs from a minor-closed class
Combinatorics, Probability and Computing
Vertices of given degree in series-parallel graphs
Random Structures & Algorithms
Vertices of degree k in random maps
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
3-connected cores in random planar graphs
Combinatorics, Probability and Computing
The maximum degree of random planar graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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Let Pn be the class of all planar graphs with n labeled vertices, and let Pn be a graph drawn uniformly at random from Pn. In this paper we study the degree sequence of Pn. We show that with probability 1 -- o(1) the number of vertices of degree k in Pn is very close to a quantity μkn that we determine explicitly, for all k ≤ c log n and an appropriate c 0. A similar statement is true for random biconnected planar graphs as well. The main tool in our analysis is a framework that allows us under certain conditions to derive universal results about the degree distribution of random graphs from general classes with structural constraints. In particular, we address so-called critical graph classes, which due to their intricate structure have posed significant technical difficulties in the past.