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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2501.18540 |
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| _version_ | 1866910913323335680 |
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| author | Di Braccio, Francesco Katsamaktsis, Kyriakos Malekshahian, Alexandru |
| author_facet | Di Braccio, Francesco Katsamaktsis, Kyriakos Malekshahian, Alexandru |
| contents | We prove that every tree of maximum degree $Δ$ with $\ell$ leaves contains paths between leaves of at least $\log_{Δ-1}((Δ-2)\ell)$ distinct lengths. This settles in a strong form a conjecture of Narins, Pokrovskiy and Szabó. We also make progress towards another conjecture of the same authors, by proving that every tree with no vertex of degree 2 and diameter at least $N$ contains $N^{2/3}/6$ distinct leaf-to-leaf path lengths between $0$ and $N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18540 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Leaf-to-leaf paths of many lengths Di Braccio, Francesco Katsamaktsis, Kyriakos Malekshahian, Alexandru Combinatorics We prove that every tree of maximum degree $Δ$ with $\ell$ leaves contains paths between leaves of at least $\log_{Δ-1}((Δ-2)\ell)$ distinct lengths. This settles in a strong form a conjecture of Narins, Pokrovskiy and Szabó. We also make progress towards another conjecture of the same authors, by proving that every tree with no vertex of degree 2 and diameter at least $N$ contains $N^{2/3}/6$ distinct leaf-to-leaf path lengths between $0$ and $N$. |
| title | Leaf-to-leaf paths of many lengths |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.18540 |