An Approximate Version of the Strong Nine Dragon Tree Conjecture
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912012145000448 |
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| author | Mies, Sebastian Moore, Benjamin |
| author_facet | Mies, Sebastian Moore, Benjamin |
| contents | We prove the Strong Nine Dragon Tree Conjecture is true if we replace the edge bound with $d + \big\lceil k \big\lfloor\frac{d-1}{k+1}\big\rfloor \big(\frac{d}{k+1} - \frac{1}{2} \big\lceil\frac{d}{k+1}\big\rceil \big)\big\rceil \leq d + \frac{k}{2} \cdot \big(\frac{d}{k+1}\big)^2$. More precisely: let $G$ be a graph, let $d$ and $k$ be positive integers and $γ(G) = \max_{H \subseteq G, v(H) \geq 2} \frac{e(H)}{v(H) - 1}$. If $γ(G) \leq k + \frac{d}{d + k + 1}$, then there is a partition of $E(G)$ into $k + 1$ forests, where in one forest every connected component has at most $d + \big\lceil k \big\lfloor\frac{d-1}{k+1}\big\rfloor \big(\frac{d}{k+1} - \frac{1}{2} \big\lceil\frac{d}{k+1}\big\rceil \big)\big\rceil$ edges. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05022 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Approximate Version of the Strong Nine Dragon Tree Conjecture Mies, Sebastian Moore, Benjamin Combinatorics We prove the Strong Nine Dragon Tree Conjecture is true if we replace the edge bound with $d + \big\lceil k \big\lfloor\frac{d-1}{k+1}\big\rfloor \big(\frac{d}{k+1} - \frac{1}{2} \big\lceil\frac{d}{k+1}\big\rceil \big)\big\rceil \leq d + \frac{k}{2} \cdot \big(\frac{d}{k+1}\big)^2$. More precisely: let $G$ be a graph, let $d$ and $k$ be positive integers and $γ(G) = \max_{H \subseteq G, v(H) \geq 2} \frac{e(H)}{v(H) - 1}$. If $γ(G) \leq k + \frac{d}{d + k + 1}$, then there is a partition of $E(G)$ into $k + 1$ forests, where in one forest every connected component has at most $d + \big\lceil k \big\lfloor\frac{d-1}{k+1}\big\rfloor \big(\frac{d}{k+1} - \frac{1}{2} \big\lceil\frac{d}{k+1}\big\rceil \big)\big\rceil$ edges. |
| title | An Approximate Version of the Strong Nine Dragon Tree Conjecture |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.05022 |