Minimum vertex degree thresholds for tiling complete 3-partite 3-graphs

TL;DR

This paper determines the minimum vertex degree needed for a 3-uniform hypergraph to contain a perfect tiling of complete 3-partite subgraphs, advancing understanding of hypergraph tiling thresholds.

Contribution

It provides an asymptotic minimum vertex degree threshold for perfect tilings of complete 3-partite 3-graphs, addressing a question posed by Mycroft.

Findings

Established the minimum vertex degree threshold for perfect $K_{a,b,c}$-tilings.

Used a lattice-based absorbing method and fractional tiling techniques.

Extended results to asymptotic conditions for 3-uniform hypergraphs.

Abstract

Given positive integers $a\leq b \leq c$, let $K_{a,b,c}$ be the complete 3-partite 3-uniform hypergraph with three parts of sizes $a,b,c$. Let $H$ be a 3-uniform hypergraph on $n$ vertices where $n$ is divisible by $a+b+c$. We asymptotically determine the minimum vertex degree of $H$ that guarantees a perfect $K_{a, b, c}$-tiling, that is, a spanning subgraph of $H$ consisting of vertex-disjoint copies of $K_{a, b, c}$. This partially answers a question of Mycroft, who proved an analogous result with respect to codegree for $r$-uniform hypergraphs for all $r\ge 3$. Our proof uses a lattice-based absorbing method, the concept of fractional tiling, and a recent result on shadows for 3-graphs.