Optimal Divisibility Conditions for Loose Hamilton Cycles in Random Hypergraphs

In the random k-uniform hypergraph H (k) n,p of order n, each possible k-tuple appears independently with probability p. A loose Hamilton cycle is a cycle of order n in which every pair of consecutive edges intersects in a single vertex. It was shown by Frieze that if p ≥ c(log n)/n2 for some absolute constant c > 0, then a.a.s. H (3) n,p contains a loose Hamilton cycle, provided that n is divisible by 4. Subsequently, Dudek and Frieze extended this result for any uniformity k ≥ 4, proving that if p ≫ (log n)/nk−1, then H n,p contains a loose Hamilton cycle, provided that n is divisible by 2(k − 1). In this paper, we improve the divisibility requirement and show that in the above results it is enough to assume that n is a multiple of k − 1, which is best possible.