A threshold result for loose Hamiltonicity in random regular uniform hypergraphs

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 ℓ-overlapping 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 and Šileikis (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.