Packing Loose Hamilton Cycles

TL;DR

This paper proves that for sufficiently high probability in a random hypergraph, one can pack nearly all edges with edge-disjoint loose Hamilton cycles, advancing understanding of cycle packings in hypergraphs.

Contribution

It establishes an asymptotically optimal bound for packing loose Hamilton cycles in random hypergraphs, improving previous bounds significantly.

Findings

High probability existence of near-complete packings of loose Hamilton cycles

Asymptotically best possible bounds up to polylogarithmic factors

Utilization and modification of online sprinkling technique

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.