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DOI: 10.1137/100790161

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Siam Journal on Discrete Mathematics 25, 4 (2011) 1615-1651

Asymptotic study of subcritical graph classes

Michael Drmota 1, Eric Fusy 2, Mihyun Kang 3, Veronika Kraus 1, Juanjo Rué 4

(2011-12-01)

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 on $n$ vertices from a subcritical graph class ${\cG}=\cup_n {\cG_n}$ satisfies asymptotically the universal behaviour $$ g_n = c n^{-5/2} \gamma^n\ (1+o(1)) $$ 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 $\cG_n$, converges (after rescaling) to a normal law as $n\to\infty$.

1:	Institut für Geometrie
	Technische Universität Wien
2:	Laboratoire d'informatique de l'école polytechnique (LIX)
	CNRS : UMR7161 – Polytechnique - X
3:	Institut für Mathematik [Berlin]
	Technische Universität Berlin
4:	Applied Mathematics IV Department
	Universitat Politécnica de Catalunya

Research team: Institut für Optimierung und Diskrete Mathematik (Math B)

Subject

:

Mathematics/Combinatorics

Keyword(s): graphs – asymptotics – limit laws

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From: Eric Fusy <>

Submitted on: Thursday, 5 July 2012 13:26:31

Updated on: Thursday, 5 July 2012 14:01:38