A threshold result for loose Hamiltonicity in random regular uniform hypergraphs

Abstract

Let G(n, r, s) denote a uniformly random r-regular s-uniform hypergraph on n vertices, where s is a fixed constant and r = r(n) may grow with n. An ℓoverlapping Hamilton cycle is a Hamilton cycle in which successive edges overlap in precisely ℓ vertices, and 1-overlapping Hamilton cycles are called loose Hamilton cycles.When r, s ≥ 3 are fixed integers, we establish a threshold result for the property of containing a loose Hamilton cycle. This partially verifies a conjecture of Dudek, Frieze, Ruciński andŠileikis (2015). In this setting, we also find the asymptotic distribution of the number of loose Hamilton cycles in G(n, r, s).Finally we prove that for ℓ = 2, . . . , s − 1 and for r growing moderately as n → ∞, the probability that G(n, r, s) has a ℓ-overlapping Hamilton cycle tends to zero.

Enumerating sparse uniform hypergraphs with given degree sequence and forbidden edges

“…see[1, Remark 6.4 and Corollary 2.3]. Using Corollary 1.2 we confirm this conjecture and extend it to the case that k and r grow sufficiently slowly with n → ∞.…”

“…When r ≥ 3 and k ≥ 2 are fixed integers, the number of Hamilton cycles in a random element of H r (k, n) has been studied by Altman et al in [1]. Let H be chosen uniformly at random from H r (k, n) and let Y be the number of loose Hamilton cycles in H. Altman et al conjectured that Proof.…”

Edge Correlations in Random Regular Hypergraphs and Applications to Subgraph Testing

“…We note that Altman, Greenhill, Isaev and Ramadurai [3] recently determined the threshold for the appearance of loose Hamilton cycles in random regular r-graphs. Their results imply that for every r ≥ 3 there exists a value d 0 (which is calculated explicitly in [3]) such that if d ≥ d 0 , then G…”

Asymptotic Enumeration of Hypergraphs by Degree Sequence

“…In the case of graphs, for instance, McKay and Wormald [18] showed that if certain asymptotic fomulae for the number of graphs with a given degree sequence hold, then the degree sequence of the random graph G(n, p) can be modelled by a sequence of independent binomial random variables. Furthermore, the above-mentioned hypergraph enumeration results were used in studying perfect matchings [7] and Hamilton cycles [1] in random regular hypergraphs, as well as enumerating linear hypergraphs with a given degree sequence [5]. Dudek, Frieze, Ruciński and Šileikis [8] gave a result that 'embeds' the Erdős-Rényi hypergraph inside the random regular hypergraph, thereby extending Kim and Vu's well known result for graphs to a denser case and also to hypergraphs.…”

“…The following theorem establishes that the space B k (n, m) can be used to model the degree sequence D k (n, m), as long as k is not too large. We use o (1) and ω(1) to denote functions of n tending to 0 and ∞, respectively, as n → ∞. We say that the convergence in the error term o() holds uniformly for all d in some set if there is a fixed upper bound g(n) on that error term, where g(n) → 0 as n → ∞, for all d in the given set.…”