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Spanning trees in random regular uniform hypergraphs

Part of: Graph theory

Published online by Cambridge University Press:  26 May 2021

Catherine Greenhill ,
Mikhail Isaev  and
Gary Liang
Catherine Greenhill*
Affiliation:
School of Mathematics and Statistics, UNSW Sydney, Sydney, NSW 2052, Australia
Mikhail Isaev
Affiliation:
School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia
Gary Liang
Affiliation:
School of Mathematics and Statistics, UNSW Sydney, Sydney, NSW 2052, Australia
*
*Corresponding author. Email: c.greenhill@unsw.edu.au
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Abstract

Let $${{\mathcal G}_{n,r,s}}$$ denote a uniformly random r-regular s-uniform hypergraph on the vertex set {1, 2, … , n}. We establish a threshold result for the existence of a spanning tree in $${{\mathcal G}_{n,r,s}}$$, restricting to n satisfying the necessary divisibility conditions. Specifically, we show that when s ≥ 5, there is a positive constant ρ(s) such that for any r ≥ 2, the probability that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree tends to 1 if r > ρ(s), and otherwise this probability tends to zero. The threshold value ρ(s) grows exponentially with s. As $${{\mathcal G}_{n,r,s}}$$ is connected with probability that tends to 1, this implies that when r ≤ ρ(s), most r-regular s-uniform hypergraphs are connected but have no spanning tree. When s = 3, 4 we prove that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree with probability that tends to 1, for any r ≥ 2. Our proof also provides the asymptotic distribution of the number of spanning trees in $${{\mathcal G}_{n,r,s}}$$ for all fixed integers r, s ≥ 2. Previously, this asymptotic distribution was only known in the trivial case of 2-regular graphs, or for cubic graphs.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C65: Hypergraphs 05C70: Factorization, matching, partitioning, covering and packing
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 1 , January 2022 , pp. 29 - 53
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

Supported by the Australian Research Council Discovery Project DP190100977.

‡

Supported by the Australian Research Council Discovery Early Career Researcher Award DE200101045.

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