The Maximum Degree of Random Planar Graphs
Michael Drmota, Omer Giménez, Marc Noy, Konstantinos D. Panagiotou, Angelika Steger · 2012
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.