An El-Zahar Type Theorem in $3$-graphs under Codegree Condition

TL;DR

This work extends El-Zahar-type tiling results to 3-graphs under a codegree regime: for any $\eta>0$ there is an $n_0$ so that, for $n\ge n_0$, a 3-graph on $n$ vertices containing a spanning collection of vertex-disjoint loose cycles $\mathcal{C}=\bigcup_i C_{n_i}$ (with total length $n$) is guaranteed to occur provided the minimum codegree satisfies $\delta_2(\mathcal{H})\ge( n+2k)/4+\eta n$, where $k$ is the number of odd cycles among the $C_{n_i}$. The result recovers and unifies prior loose Hamilton cycle and loose cycle-factor results by Kühn–Osthus and Mycroft, respectively. The proof combines the (degree form of the) weak hypergraph regularity lemma with a transversal blow-up lemma, enabling a reduction to a reduced hypergraph tiling and then lifting the embedding to the original graph. The finding is asymptotically tight and broadens the scope of El-Zahar-type tilings from graphs to 3-graphs under a precise codegree condition, with potential extensions to larger uniformities and different tiling families.

Abstract

A $3$-uniform loose cycle, denoted by $C_t$, is a $3$-graph on $t$ vertices whose vertices can be arranged cyclically so that each hyperedge consists of three consecutive vertices, and any two consecutive hyperedges share exactly one vertex. The length of $C_t$ is the number of its hyperedges. We prove that for any $η>0$, there exists an $n_0=n_0(η)$ such that for any $n\geq n_0$ the following holds. Let $\mathcal{C}$ be a $3$-graph consisting of vertex-disjoint loose cycles $C_{n_1}, C_{n_2}, \ldots, C_{n_r}$ such that $\sum_{i=1}^{r}n_i=n$. Let $k$ be the number of loose cycles with odd lengths in $\mathcal{C}$. If $\mathcal{H}$ is a $3$-graph on $n$ vertices with minimum codegree at least $(n+2k)/4+ηn$, then $\mathcal{H}$ contains $\mathcal{C}$ as a spanning subhypergraph. The degree condition is approximately tight. This generalizes the result of Kühn and Osthus for loose Hamilton cycle and the result of Mycroft for loose cycle factors in $3$-graphs. Our proof relies on the regularity lemma and a transversal blow-up lemma recently developed by the first author and Staden.

Theorems & Definitions (50)

• Conjecture 1.1: el1984circuits, El-Zahar's conjecture
• Theorem 1.2: kuhn2006loose
• Theorem 1.3: Czy2016JGT
• Theorem 1.4: Main result
• Proposition 1.5
• proof
• Definition 2.1
• proof
• Lemma 2.3: kuhn2006loose, Proposition 4.1
• Lemma 2.4