Almost all maps are asymmetric
Journal of Combinatorial Theory Series B
A pattern of asymptotic vertex valency distributions in planar maps
Journal of Combinatorial Theory Series B
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
The Size of the Largest Components in Random Planar Maps
SIAM Journal on Discrete Mathematics
Boltzmann Samplers for the Random Generation of Combinatorial Structures
Combinatorics, Probability and Computing
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
Optimal succinct representations of planar maps
Proceedings of the twenty-second annual symposium on Computational geometry
On properties of random dissections and triangulations
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Analytic Combinatorics
On the degree distribution of random planar graphs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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This work is devoted to the study of the typical structure of a random map. Maps are planar graphs embedded in the plane. We investigate the degree sequences of random maps from families of a certain type, which, among others, includes fundamental map classes like those of biconnected maps, 3-connected maps, and triangulations. In particular, we develop a general framework that allows us to derive relations and exact asymptotic expressions for the expected number of vertices of degree k in random maps from these classes, and also provide accompanying large deviation statements. Extending the work of Gao and Wormald (Combinatorica, 2003) on random general maps, we obtain as results of our framework precise information about the number of vertices of degree k in random biconnected, 3-connected, loopless, and bridgeless maps.