Let G n,r,s denote a uniformly random r-regular s-uniform hypergraph on the vertex set {1, 2, . . . , n}. We analyse the asymptotic distribution of Y G , the number of spanning trees in G, when r, s 2 are fixed constants, (r, s) = (2, 2), and the necessary divisibility conditions hold. Greenhill, Kwan and Wind (2014) investigated the graph case (s = 2), providing an asymptotic formula for the expected value of Y G for any fixed r 3, which was previously only known up to a constant factor. They also found the asymptotic distribution of Y G for random cubic graphs (r = 3), and made a conjecture for arbitrary r 4. Here we prove this conjecture and extend the analysis to hypergraphs. When s 5 we prove a threshold result for the property that G n,r,s contains a spanning tree. We also calculate the asymptotic distribution of Y G for all fixed integers r, s 2 such that the probability that G n,r,s has a spanning tree tends to one as n grows.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.