We investigate the asymptotic structure of a random perfect graph Pn​
sampled uniformly from the perfect graphs on vertex set {1,…,n}. Our
approach is based on the result of Pr\"omel and Steger that almost all perfect
graphs are generalised split graphs, together with a method to generate such
graphs almost uniformly.
We show that the distribution of the maximum of the stability number
α(Pn​) and clique number ω(Pn​) is close to a concentrated
distribution L(n) which plays an important role in our generation method. We
also prove that the probability that Pn​ contains any given graph H as an
induced subgraph is asymptotically 0 or 21​ or 1. Further we show
that almost all perfect graphs are 2-clique-colourable, improving a result of
Bacs\'o et al from 2004; they are almost all Hamiltonian; they almost all have
connectivity κ(Pn​) equal to their minimum degree; they are almost all
in class one (edge-colourable using Δ colours, where Δ is the
maximum degree); and a sequence of independently and uniformly sampled perfect
graphs of increasing size converges almost surely to the graphon WP​(x,y)=21​(1[x≤1/2]+1[y≤1/2])