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

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.