Codegree threshold for tiling $k$-graphs with two edges sharing exactly $\ell$ vertices

TL;DR

This paper determines the exact codegree threshold for tiling $k$-graphs with two edges sharing exactly $ ext{ extbackslash}ell$ vertices, solving a problem posed by Han and Zhao for all relevant parameters.

Contribution

The paper establishes the exact value of the codegree threshold for tiling with two edges sharing $ ext{ extbackslash}ell$ vertices for a wide range of parameters, completing a previously open problem.

Findings

Exact threshold $t_{k-1}(n, ext{ extbackslash}mathcal{Y}_{k, ext{ extbackslash}ell})=\frac{n}{2k- ext{ extbackslash}ell}$ for $k extbackslash}geq 3$ and $1 extbackslash}leq extbackslash}ell extbackslash}leq k-2$.

Solved the open problem posed by Han and Zhao.

Unified previous results to fully determine the threshold.

Abstract

Given integer $k$ and a $k$-graph $F$, let $t_{k-1}(n,F)$ be the minimum integer $t$ such that every $k$-graph $H$ on $n$ vertices with codegree at least $t$ contains an $F$-factor. For integers $k\geq3$ and $0\leq\ell\leq k-1$, let $\mathcal{Y}_{k,\ell}$ be a $k$-graph with two edges that shares exactly $\ell$ vertices. Han and Zhao (JCTA, 2015) asked the following question: For all $k\ge 3$, $0\le \ell\le k-1$ and sufficiently large $n$ divisible by $2k-\ell$, determine the exact value of $t_{k-1}(n,\mathcal{Y}_{k,\ell})$. In this paper, we show that $t_{k-1}(n,\mathcal{Y}_{k,\ell})=\frac{n}{2k-\ell}$ for $k\geq3$ and $1\leq\ell\leq k-2$, combining with two previously known results of R\"{o}dl, Ruci\'{n}ski and Szemer\'{e}di {(JCTA, 2009)} and Gao, Han and Zhao (arXiv, 2016), the question of Han and Zhao is solved completely.