### Packing Loose Hamilton Cycles [article]

Asaf Ferber, Kyle Luh, Daniel Montealegre, Oanh Nguyen
2016 arXiv   pre-print
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
more » ... -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( 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≥^Cn/n^k-1, a random k-uniform hypergraph H^k_n,p with high probability contains N:=(1-o(1))nkp/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.