On the probability of planarity of a random graph near the critical point

Abstract

Let G(n, M ) be the uniform random graph with n vertices and M edges. Erdős and Rényi (1960) conjectured that the limiting probability) is planar} exists and is a constant strictly between 0 and 1. Luczak, Pittel and Wierman (1994) proved this conjecture and Janson, Luczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact limiting probability of a random graph being planar near the critical point M = n/2. For each λ, we find an exact analytic expression for(1 + λn −1/3 ) is planar .In particular, we obtain p(0) ≈ 0.99780. We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of G(n, n 2 ) being seriesparallel converges to 0.98003. For the sake of completeness and exposition we reprove in a concise way several basic properties we need of a random graph near the critical point.We dedicate this paper to the memory of Philippe Flajolet.

Spanning trees in random series-parallel graphs

“…Before moving to the computation of g k and g k , let us mention that similar arguments give estimates for [x n ]C k (x). Indeed, using the argument of Lemma 3 in [35] (or mutatis mutandis the previous arguments for C k (x)) one gets the following upper and lower bounds for [x n ]C k (x):…”

“…Proof. The proof is reminiscent to the proof of Lemma 3 in [35]. The EGF on the left hand side of Equation (28) can be written in the following way:…”

The genus of the Erdős‐Rényi random graph and the fragile genus property

“…Taking into account later work by Rödl and Thomas [31] (which deals with a substantially wider range for p), these show that g(n, m) = (1 + o(1)) m 6 whp when m = Θ(n 2 ) and that g(n, m) = (1 + o(1)) jm 2(j+2) whp when n 1+ 1 j+1 ≪ m ≪ n 1+ 1 j for j ∈ N. Separately, important work has also been carried out to determine the probability that G(n, m) is planar (i.e., has zero genus) when m is comparatively small. In particular, it is now well-known that G(n, m) is planar whp when m < n 2 − ( n 2∕3 ) (see [27]) and that lim inf P[G(n, m) is planar] > 0 when m = n 2 + O ( n 2∕3 ) (see [27] and [30]). For other interesting results in this area, see also [13] and [17].…”

“…Separately, important work has also been carried out to determine the probability that G ( n , m ) is planar (i.e., has zero genus) when m is comparatively small. In particular, it is now well‐known that G ( n , m ) is planar whp when (see ) and that when (see and ). For other interesting results in this area, see also and .…”

Random Planar Maps and Graphs with Minimum Degree Two and Three

“…To this end we weight multigraphs according to the number of loops and edges of each multiplicity. We remark that the concepts of core and kernel of a graph are instrumental in the classical theory of random graphs [14,17].…”