Asymptotic Study of Subcritical Graph Classes

Michael Drmota, Éric Fusy, Mihyun Kang, Veronika Kraus, Juanjo Rué
2011 SIAM Journal on Discrete Mathematics  
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 both in 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
more » ... n n vertices from a subcritical graph class G=∪_n G_n satisfies asymptotically the universal behaviour g_n = c n^-5/2γ^n (1+o(1)) for computable constants c,γ, e.g. γ≈ 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→∞.
doi:10.1137/100790161 fatcat:eh2dxsvv7ngdbkyvnt267lyvsi