The Maximum Degree of Random Planar Graphs

Abstract

Let P n 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 P n 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 P n . 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 P n .

The maximal degree in a Poisson–Delaunay graph

“…Our theorem is rather classical in the sense that similar results have already been established in the context of random combinatorial graphs [5,9,15,16,17,23,24] and Gilbert graph [26,Th 6.6]. Besides, Anderson [1] proved that the maximum of n independent and identically distributed random variables is concentrated, with high probability as n goes to infinity, on two consecutive integers for a wide class of discrete random variables.…”

“…The maximal degree of random combinatorial graphs has been extensively investigated, see e.g. [5,9,15,16,17,23,24]. Much less has been done when the vertices are given by a point process and the edges built according to geometric constraints.…”

Maximum degree in minor-closed classes of graphs

“…More recently this result has been strengthened using subtle analytic and probabilistic methods [5], by showing the existence of a computable constant c such that ∆ n log n → c in probability.…”

The Evolution of Random Graphs on Surfaces

“…, n}) are now known. For example, asymptotic results have been obtained for the probability that P (n) will contain given components and subgraphs [20,28], for the number of vertices of given degree [13], and for the size of the maximum degree and largest face [12,27]. In addition, clever algorithms for generating and sampling planar graphs have been designed [6,15], and random planar maps have been studied [14,16,17].…”