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 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.