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 co...
| Autores: | , , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2011 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/104308 |
| Acceso en línea: | https://hdl.handle.net/2117/104308 https://dx.doi.org/10.1137/100790161 |
| Access Level: | acceso abierto |
| Palabra clave: | Graph theory Graph classes Grafs, Teoria dels Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs |
| Sumario: | 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 $n\to\infty$. |
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