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Degree sequences of sufficiently dense random uniform hypergraphs

Part of: Combinatorics Graph theory

Published online by Cambridge University Press:  15 August 2022

Catherine Greenhill Open the ORCID record for Catherine Greenhill [Opens in a new window] ,
Mikhail Isaev Open the ORCID record for Mikhail Isaev [Opens in a new window] ,
Tamás Makai Open the ORCID record for Tamás Makai [Opens in a new window]  and
Brendan D. McKay Open the ORCID record for Brendan D. McKay [Opens in a new window]
Catherine Greenhill
Affiliation:
School of Mathematics and Statistics, UNSW Sydney, NSW 2052, Australia
Mikhail Isaev
Affiliation:
School of Mathematics, Monash University, VIC 3800, Australia
Tamás Makai
Affiliation:
School of Mathematics and Statistics, UNSW Sydney, NSW 2052, Australia
Brendan D. McKay*
Affiliation:
School of Computing, Australian National University, Canberra, ACT 2601, Australia
*
*Corresponding author. Email: brendan.mckay@anu.edu.au
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Abstract

We find an asymptotic enumeration formula for the number of simple $r$ -uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations. We give sufficient conditions on the degree sequence which guarantee existence of a solution to this system. Furthermore, we solve the system and give an explicit asymptotic formula when the degree sequence is close to regular. This allows us to establish several properties of the degree sequence of a random $r$ -uniform hypergraph with a given number of edges. More specifically, we compare the degree sequence of a random $r$ -uniform hypergraph with a given number edges to certain models involving sequences of binomial or hypergeometric random variables conditioned on their sum.

Keywords

random hypergraphdegree sequenceasymptotic enumeration

MSC classification

Primary: 05A16: Asymptotic enumeration
Secondary: 05C80: Random graphs 05C65: Hypergraphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 32 , Issue 2 , March 2023 , pp. 183 - 224
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

Research supported by the Australian Research Council, Discovery Project DP190100977.

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