Analytic combinatorics of non-crossing configurations
Discrete Mathematics - Special issue on selected papers in honor of Henry W. Gould
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
On properties of random dissections and triangulations
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Analytic Combinatorics
On properties of random dissections and triangulations
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
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
The degree sequence of random graphs from subcritical classes†
Combinatorics, Probability and Computing
Maximal biconnected subgraphs of random planar graphs
ACM Transactions on Algorithms (TALG)
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
On the degree distribution of random planar graphs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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In the past decades the Gn,p model of random graphs, introduced by Erdős and Rényi in the 60's, has led to numerous beautiful and deep theorems. A key feature that is used in basically all proofs is that edges in Gn,p appear independently. The independence of the edges allows, for example, to obtain extremely tight bounds on the number of edges of Gn,p and its degree sequence by straightforward applications of Chernoff bounds. This situation changes dramatically if one considers graph classes with structural side constraints. For example, in a random planar graph Rn (a graph drawn uniformly at random from the class of all labeled planar graphs on n vertices) the edges are obviously far from being independent. Consequently, so far basically all results about properties of random graphs with structural side constraints rely on completely different methods, mostly from analytic combinatorics. In this paper we show that recent progress in the construction of so-called Boltzmann samplers by Duchon, Flajolet, Louchard, and Schaeffer [3] and Fusy [6] can be used to reduce the study of degree sequences and subgraph counts to properties of sequences of independent and identically distributed random variables -- to which we can then again apply Chernoff bounds to obtain extremely tight results. We elaborate our ideas by studying random dissections and triangulations of a labeled convex n-gon. For both we obtain the degree sequence and the number of induced copies of given fixed graphs. The degree sequence for triangulations was already obtained previously by Gao and Wormald [8] using deep methods from analytic combinatorics. We do, however, get better bounds for the tails of the probability distribution.