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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.20535 |
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| _version_ | 1866910948979113984 |
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| author | Cheng, Yangyang Rao, Mengjiao Wang, Guanghui Zhao, Yuqi |
| author_facet | Cheng, Yangyang Rao, Mengjiao Wang, Guanghui Zhao, Yuqi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20535 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An El-Zahar Type Theorem in $3$-graphs under Codegree Condition Cheng, Yangyang Rao, Mengjiao Wang, Guanghui Zhao, Yuqi Combinatorics 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. |
| title | An El-Zahar Type Theorem in $3$-graphs under Codegree Condition |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.20535 |