Asymptotic enumeration of sparse 2-connected graphs

DOI10.1002/RSA.20415zbMath1273.05127arXiv1010.2516OpenAlexW2592011129MaRDI QIDQ2856579

Cristiane M. Sato, Graeme Kemkes, Nicholas C. Wormald

Publication date: 29 October 2013

Published in: Unnamed Author (Search for Journal in Brave)

Abstract: We determine an asymptotic formula for the number of labelled 2-connected (simple) graphs on $n$ vertices and $m$ edges, provided that $m-n oinfty$ and $m=O(nlog n)$ as $n oinfty$. This is the entire range of $m$ not covered by previous results. The proof involves determining properties of the core and kernel of random graphs with minimum degree at least 2. The case of 2-edge-connectedness is treated similarly. We also obtain formulae for the number of 2-connected graphs with given degree sequence for most (`typical') sequences. Our main result solves a problem of Wright from 1983.

Full work available at URL: https://arxiv.org/abs/1010.2516

zbMATH Keywords

random graphsasymptotic enumeration2-connected graphs

Mathematics Subject Classification ID

Random graphs (graph-theoretic aspects) (05C80) Enumeration in graph theory (05C30) Asymptotic enumeration (05A16) Connectivity (05C40) Density (toughness, etc.) (05C42)

Cites Work

• Unnamed Item
• The exponential generating function of labelled blocks
• The asymptotic number of labeled graphs with \(n\) vertices, \(q\) edges, and no isolated vertices
• Counting connected graphs inside-out
• Asymptotic enumeration of sparse graphs with a minimum degree constraint
• Forbidden subgraphs in connected graphs
• On the number of graphs with a fixed number of vertices, edges, and isolated vertices
• On the Enumeration of Mayer Cluster Integrals
• On the strength of connectedness of a random graph
• The number of connected sparsely edged graphs. IV large nonseparable graphs
• The asymptotic number of labeled connected graphs with a given number of vertices and edges
• The number of connected sparsely edged graphs. II. Smooth graphs and blocks

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