NDLI: Asymptotic Study of Subcritical Graph Classes

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Fusy, ric Kraus, Veronika Ru, Juanjo Kang, Mihyun Drmota, Michael
Copyright Year	2011
Abstract	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 ${\mathcal{G}}=\cup_n {\mathcal{G}_n}$ satisfies asymptotically the universal behavior $g_n = c \!n^{-5/2} \!\gamma^n \! (1+o(1))$ for computable constants $c,\gamma$, e.g., $\gamma\approx 9.38527$ for unlabelled series-parallel graphs, and that the number of vertices of degree k (k fixed) in a graph chosen uniformly at random from $\mathcal{G}_n$ converges (after rescaling) to a normal law as $n\to\infty$.
Starting Page	1615
Ending Page	1651
Page Count	37
File Format	PDF
ISSN	08954801
DOI	10.1137/100790161
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	4
Volume Number	25
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-12-01
Access Restriction	Subscribed
Subject Keyword	Enumeration in graph theory random graph asymptotic enumeration analytic combinatorics limit law Asymptotic enumeration
Content Type	Text
Resource Type	Article
Subject	Mathematics

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