Boltzmann Samplers for the Random Generation of Combinatorial Structures
Combinatorics, Probability and Computing
Journal of Combinatorial Theory Series B
On the maximum degree of a random planar graph
Combinatorics, Probability and Computing
Analytic Combinatorics
Random graphs from a minor-closed class
Combinatorics, Probability and Computing
Degree distribution in random planar graphs
Journal of Combinatorial Theory Series A
On the degree distribution of random planar graphs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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Let Pn denote a graph drawn uniformly at random from the class of all simple planar graphs with n vertices. We show that the maximum degree of a vertex in Pn is with probability 1 − o(1) asymptotically equal to c log n, where c ≈ 2.529 is determined explicitly. A similar result is also true for random 2-connected planar graphs. Our analysis combines two orthogonal methods that complement each other. First, in order to obtain the upper bound, we resort to exact methods, i.e., to generating functions and analytic combinatorics. This allows us to obtain fairly precise asymptotic estimates for the expected number of vertices of any given degree in Pn. On the other hand, for the lower bound we use Boltzmann sampling. In particular, by tracing the execution of an adequate algorithm that generates a random planar graph, we are able to explicitly find vertices of sufficiently high degree in Pn.