Codegree conditions for tilling balanced complete $3$-partite $3$-graphs and generalized 4-cycles

TL;DR

This paper advances the understanding of minimum codegree thresholds for perfect tilings in 3-uniform hypergraphs, improving error bounds for specific graphs and confirming Mycroft's conjecture for a generalized 4-cycle.

Contribution

It refines the error term in minimum codegree thresholds for perfect tilings of complete 3-partite 3-graphs and verifies Mycroft's conjecture for the generalized 4-cycle.

Findings

Improved the error term to sub-linear for $K^3(m)$.

Proved the tightness of the sub-linear error term for $K^3(2)$.

Confirmed Mycroft's conjecture for the generalized 4-cycle $C_4^3$.

Abstract

Given two $k$-graphs $F$ and $H$, a perfect $F$-tiling (also called an $F$-factor) in $H$ is a set of vertex disjoint copies of $F$ that together cover the vertex set of $H$. Let $t_{k-1}(n, F)$ be the smallest integer $t$ such that every $k$-graph $H$ on $n$ vertices with minimum codegree at least $t$ contains a perfect $F$-tiling. Mycroft (JCTA, 2016) determined the asymptotic values of $t_{k-1}(n, F)$ for $k$-partite $k$-graphs $F$. Mycroft also conjectured that the error terms $o(n)$ in $t_{k-1}(n, F)$ can be replaced by a constant that depends only on $F$. In this paper, we improve the error term of Mycroft's result to a sub-linear term when $F=K^3(m)$, the complete $3$-partite $3$-graph with each part of size $m$. We also show that the sub-linear term is tight for $K^3(2)$, {the result also provides another family of counterexamples of Mycroft's conjecture (Gao, Han, Zhao (arXiv, 2016) gave a family of counterexamples when $H$ is a $k$-partite $k$-graph with some restrictions.)} Finally, we show that Mycroft's conjecture holds for generalized 4-cycle $C_4^3$ (the 3-graph on six vertices and four distinct edges $A, B, C, D$ with $A\cup B= C\cup D$ and $A\cap B=C\cap D=\emptyset$), i.e. we determine the exact value of $t_2(n, C_4^3)$.