Asymptotic enumeration of sparse 2-connected graphs

Authors

  • Graeme Kemkes,

    1. Department of Mathematics, Ryerson University, Toronto, Ontario, Canada M5B 2K3
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    • This author's research was primarily carried out while at the University of California at San Diego under an NSERC postdoctoral award.

  • Cristiane M. Sato,

    Corresponding author
    1. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
    • Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
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  • Nicholas Wormald

    1. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
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    • Supported by the Canada Research Chairs Program and NSERC.


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 and m = O(nlog n) as n. 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. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013

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