Packing Loose Hamilton Cycles

Abstract: A subset $C$ of edges in a $k$-uniform hypergraph $H$ is a \emph{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 $e\in \binom{[n]}{k}$ 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 \emph{packing problem}. While it is known that the threshold probability for the appearance of a loose Hamilton cycle in $H^k_{n,p}$ is $p=\Theta\left(\frac{\log n}{n^{k-1}}\right)$, the best known bounds for the packing problem are around $p=\text{polylog}(n)/n$. Here we make substantial progress and prove the following asymptotically (up to a polylog$(n)$ factor) best possible result: For $p\geq \log^{C}n/n^{k-1}$, a random $k$-uniform hypergraph $H^k_{n,p}$ with high probability contains $N:=(1-o(1))\frac{\binom{n}{k}p}{n/(k-1)}$ edge-disjoint loose Hamilton cycles. Our proof utilizes and modifies the idea of "online sprinkling" recently introduced by Vu and the first author.