Random planar graphs

Abstract

We study various properties of the random planar graph R n , drawn uniformly at random from the class P n of all simple planar graphs on n labelled vertices. In particular, we show that the probability that R n is connected is bounded away from 0 and from 1. We also show for example that each positive integer k, with high probability R n has linearly many vertices of a given degree, in each embedding R n has linearly many faces of a given size, and R n has exponentially many automorphisms.

Asymptotic enumeration and limit laws of planar graphs

“…, n}, where n > h. Let W ⊂ V (G) with |W | = h, and let r W denote the least element in W . Following [12], we say that H appears at W in G if (a) the increasing bijection from {1, . .…”

“…The reason for the strict inequality γ < γ u is that, contrary to what happens for unrestricted graphs, a planar graph has with high probability an exponential number of automorphisms [12]. The best upper bound obtained so far is γ u < 30.06.…”

“…1/n , and that γ < γ u , where γ is as in Theorem 1 (see [12]). The reason for the strict inequality γ < γ u is that, contrary to what happens for unrestricted graphs, a planar graph has with high probability an exponential number of automorphisms [12].…”

How randomized search heuristics find maximum cliques in planar graphs

“…Let H be the graph described in the proof of Part 2 of Lemma 8 on 6 + 4 vertices. Similar to [10,Proposition 4.4] the random planar graph on n vertices contains at least one subgraph H d 1 log n/ log log n with a failure probability of at most n −d /3, for a constant d1 > 0 depending on d. Moreover, similar to [10, Proposition 4.5] the random planar graph on n vertices consists of vertices with a degree of at most d2 log n only with a failure probability of at most n −d /3, for a constant d2 > 0 depending on d. Let us assume that all this holds in the following.…”

“…Simple expected (w.r.t. the random input) constant time algorithms find a maximum clique in Gn [10]. However, real-world instances are generally not as well-formed as random ones.…”