Packing k-partite k-uniform hypergraphs

TL;DR

This paper determines the asymptotic minimum codegree conditions needed for perfect packings of k-partite k-uniform hypergraphs, extending known results from graphs to hypergraphs and solving longstanding open problems.

Contribution

It provides the first asymptotic thresholds for perfect packings of complete k-partite k-graphs in hypergraphs, including loose cycles, for all k ≥ 3.

Findings

Established asymptotic minimum codegree thresholds for perfect H-packings.

Solved the problem for all complete k-partite k-graphs and a broad class of others.

Extended classical graph packing results to hypergraphs.

Abstract

Let $G$ and $H$ be $k$-graphs ($k$-uniform hypergraphs); then a perfect $H$-packing in $G$ is a collection of vertex-disjoint copies of $H$ in $G$ which together cover every vertex of $G$. For any fixed $H$ let $\delta(H, n)$ be the minimum $\delta$ such that any $k$-graph $G$ on $n$ vertices with minimum codegree $\delta(G) \geq \delta$ contains a perfect $H$-packing. The problem of determining $\delta(H, n)$ has been widely studied for graphs (i.e. $2$-graphs), but little is known for $k \geq 3$. Here we determine the asymptotic value of $\delta(H, n)$ for all complete $k$-partite $k$-graphs $H$, as well as a wide class of other $k$-partite $k$-graphs. In particular, these results provide an asymptotic solution to a question of R\"odl and Ruci\'nski on the value of $\delta(H, n)$ when $H$ is a loose cycle. We also determine asymptotically the codegree threshold needed to guarantee an $H$-packing covering all but a constant number of vertices of $G$ for any complete $k$-partite $k$-graph $H$.