ASYMPTOTIC STUDY OF SUBCRITICAL GRAPH CLASSES

We present a unified general method for the asymptotic study of graphs from the so-called subcritical graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works in both the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number g(n)/n! (resp., g(n)) of labelled (resp., unlabelled) graphs on n vertices from a subcritical graph class g = boolean OR(n)g(n) satisfies asymptotically the universal behavior g(n) = c n(-5/2) gamma(n) (1 + o(1)) for computable constants c, gamma, e.g., gamma approximate to 9.38527 for unlabelled series-parallel graphs, and that the number of vertices of degree k (k fixed) in a graph chosen uniformly at random from g(n) converges (after rescaling) to a normal law as n -> infinity.