Packing Loose Hamilton Cycles

Abstract

Abstract. 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 for the appearance of a loose Hamilton cycle in H k n,p is p = Θ log n 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 k n,p with high probability contains N :edgedisjoint loose Hamilton cycles.Our proof utilizes and modifies the idea of "online sprinkling" recently introduced by Vu and the first author.