Packing Loose Hamilton Cycles

A subset C of edges in a k-uniform hypergraph H is a loose Hamilton cycle if C covers all the vertices of H and there exists a cyclic ordering of these vertices such that the edges in C are segments of that order and such that every two consecutive edges share exactly one vertex. The binomial random k-uniform hypergraph H ^k _n,p has vertex set [n] and an edge set E obtained by adding each k-tuple e ∈ (^[n] _k) to E with probability p, independently at random. Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all but o(|E|) edges, referred to as the packing problem. While it is known that the threshold probability of the appearance of a loose Hamilton cycle in H ^k _n,p is {equation presented} the best known bounds for the packing problem are around p = polylog(n)/n. Here we make substantial progress and prove the following asymptotically (up to a polylog(n) factor) best possible result: for {equation presented}, a random k-uniform hypergraph H ^k _n,p with high probability contains {equation presented} edge-disjoint loose Hamilton cycles. Our proof utilizes and modifies the idea of 'online sprinkling' recently introduced by Vu and the first author.