Cycle-factors in oriented graphs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929262158675968 |
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| author | Wang, Zhilan Yan, Jin Zhang, Jie |
| author_facet | Wang, Zhilan Yan, Jin Zhang, Jie |
| contents | Let $k$ be a positive integer. A $k$-cycle-factor of an oriented graph is a set of disjoint cycles of length $k$ that covers all vertices of the graph. In this paper, we prove that there exists a positive constant $c$ such that for $n$ sufficiently large, any oriented graph on $n$ vertices with both minimum out-degree and minimum in-degree at least $(1/2-c)n$ contains a $k$-cycle-factor for any $k\geq4$. Additionally, under the same hypotheses, we also show that for any sequence $n_1, \ldots, n_t$ with $\sum^t_{i=1}n_i=n$ and the number of the $n_i$ equal to $3$ is $αn$, where $α$ is any real number with $0<α<1/3$, the oriented graph contains $t$ disjoint cycles of lengths $n_1, \ldots, n_t$. This conclusion is the best possible in some sense and refines a result of Keevash and Sudakov. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05077 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cycle-factors in oriented graphs Wang, Zhilan Yan, Jin Zhang, Jie Combinatorics 05C70, 05C20, 05C38 Let $k$ be a positive integer. A $k$-cycle-factor of an oriented graph is a set of disjoint cycles of length $k$ that covers all vertices of the graph. In this paper, we prove that there exists a positive constant $c$ such that for $n$ sufficiently large, any oriented graph on $n$ vertices with both minimum out-degree and minimum in-degree at least $(1/2-c)n$ contains a $k$-cycle-factor for any $k\geq4$. Additionally, under the same hypotheses, we also show that for any sequence $n_1, \ldots, n_t$ with $\sum^t_{i=1}n_i=n$ and the number of the $n_i$ equal to $3$ is $αn$, where $α$ is any real number with $0<α<1/3$, the oriented graph contains $t$ disjoint cycles of lengths $n_1, \ldots, n_t$. This conclusion is the best possible in some sense and refines a result of Keevash and Sudakov. |
| title | Cycle-factors in oriented graphs |
| topic | Combinatorics 05C70, 05C20, 05C38 |
| url | https://arxiv.org/abs/2402.05077 |