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

TL;DR

This paper proves a near-tight minimum codegree condition for spanning loose cycle factors in 3-uniform hypergraphs, generalizing previous results and employing advanced regularity and blow-up lemmas.

Contribution

It extends El-Zahar type theorems to 3-graphs with codegree conditions, covering spanning loose cycle factors with new proof techniques.

Findings

Minimum codegree threshold for spanning loose cycle factors established

Generalizes previous results for loose Hamilton cycles and cycle factors

Uses regularity lemma and transversal blow-up lemma in proof

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 $\eta>0$, there exists an $n_0=n_0(\eta)$ 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 $(n+2k)/4+\eta n$, then $\mathcal{H}$ contains $\mathcal{C}$ as a spanning subhypergraph. The degree condition is approximately tight. This generalizes the result of K\"{u}hn 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.