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-n,p(k) has vertex set [n] and an edge set E obtained by adding each k-tuple e is an element of (([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(vertical bar E vertical bar) 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-n,p(k) is p = Theta( logn/n(k-1)) , 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 p >= log(C) n/n(k-1), a random k-uniform hypergraph H-n,p(k) with high probability contains N := (1-o(1)) ((n)(k))p/n/(k-1) edge-disjoint loose Hamilton cycles. Our proof utilizes and modifies the idea of 'online sprinkling' recently introduced by Vu and the first author.