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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2509.18832 |
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| _version_ | 1866918150348472320 |
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| author | Lo, Allan |
| author_facet | Lo, Allan |
| contents | A typical Dirac-type problem in extremal graph theory is to determine the minimum degree threshold for a graph $G$ to have a spanning subgraph $H$, e.g. the Dirac theorem. A natural following up problem would be to seek an $H$-factor, which a spanning set of vertex-disjoint copies of $H$. In this short note, we present a method of obtaining an upper bound on the minimum degree threshold for an $H$-factor from one for finding a spanning copy of $H$.
As an application, we proved that, for all $\varepsilon>0$ and $\ell$ sufficiently large, any oriented graph $G$ on $\ell m$ vertices with minimum semi-degree $δ^0(G) \ge (3/8+ \varepsilon) k \ell$ contains a $C_\ell$-factor, where $C_\ell$ is an arbitrary orientation of a cycle on $\ell$ vertices. This improves a result of Wang, Yan and Zhang. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18832 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From finding a spanning subgraph $H$ to an $H$-factor Lo, Allan Combinatorics A typical Dirac-type problem in extremal graph theory is to determine the minimum degree threshold for a graph $G$ to have a spanning subgraph $H$, e.g. the Dirac theorem. A natural following up problem would be to seek an $H$-factor, which a spanning set of vertex-disjoint copies of $H$. In this short note, we present a method of obtaining an upper bound on the minimum degree threshold for an $H$-factor from one for finding a spanning copy of $H$. As an application, we proved that, for all $\varepsilon>0$ and $\ell$ sufficiently large, any oriented graph $G$ on $\ell m$ vertices with minimum semi-degree $δ^0(G) \ge (3/8+ \varepsilon) k \ell$ contains a $C_\ell$-factor, where $C_\ell$ is an arbitrary orientation of a cycle on $\ell$ vertices. This improves a result of Wang, Yan and Zhang. |
| title | From finding a spanning subgraph $H$ to an $H$-factor |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.18832 |