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 onn vertices, where s is a fixed constant and r = r(n) may grow with n. An `-overlapping Hamilton cycle is a Hamilton cycle in which successive edges overlapin precisely ` vertices, and 1-overlapping Hamilton cycles are called loose Hamiltoncycles.When r, s ≥ 3 are fixed integers, we establish a threshold result for the prop-erty of containing a loose Hamilton cycle. This partially verifies a conjectureof Dudek, Frieze, Ruciński and Šileikis (2015). In this setting, we also find theasymptotic distribution and expected value of the number of loose Hamilton cyclesin G(n, r, s).Finally we prove that for ` = 2, . . . , s − 1 and for r growing moderately asn → ∞, the probability that G(n, r, s) has a `-overlapping Hamilton cycle tendsto zero.∗Supported by the Australian Research Council grant DP140101519.