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

Consider the uniform random graph $G(n,M)$ with $n$ vertices and $M$ edges. Erd\H{o}s and R\'enyi (1960) conjectured that the limit \lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} exists and is a constant strictly between 0 and 1. \L uczak, Pittel and Wierman (1994) proved this conjecture and Janson, \L uczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact probability of a random graph being planar near the critical point $M=n/2$. For each $\lambda$, we find an exact analytic expression for $$p(\lambda) = \lim_{n \to \infty} \Pr{G(n,\textstyle{n\over 2}(1+\lambda n^{-1/3})) is planar}.$$ In particular, we obtain $p(0) \approx 0.99780$. We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of $G(n,\textstyle{n\over 2})$ being series-parallel 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.Comment: 10 pages, 1 figur

Degree distribution in random planar graphs

We prove that for each $k\ge0$, the probability that a root vertex in a random planar graph has degree $k$ tends to a computable constant $d_k$, so that the expected number of vertices of degree $k$ is asymptotically $d_k n$, and moreover that $\sum_k d_k =1$. The proof uses the tools developed by Gimenez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function $p(w)=\sum_k d_k w^k$. From this we can compute the $d_k$ to any degree of accuracy, and derive the asymptotic estimate $d_k \sim c\cdot k^{-1/2} q^k$ for large values of $k$, where $q \approx 0.67$ is a constant defined analytically