Pancyclicity in hypergraphs with large uniformity
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918005685878784 |
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| author | Bailey, Teegan Hollars, Isaiah Li, Yupei Luo, Ruth |
| author_facet | Bailey, Teegan Hollars, Isaiah Li, Yupei Luo, Ruth |
| contents | A Berge cycle of length $\ell$ in a hypergraph $\mathcal{H}$ is a sequence of alternating vertices and edges $v_0e_0v_1e_1...v_\ell e_\ell v_0$ such that $\{v_i,v_{i+1}\}\subseteq e_i$ for all $i$, with indices taken modulo $\ell$.
For $n$ sufficiently large and $r\geq \lfloor\frac{n-1}{2}\rfloor-1$ we prove exact minimum degree conditions for an $n$-vertex, $r$-uniform hypergraph to contain Berge cycles of every length between $2$ and $n$. In conjunction with previous work, this provides sharp Dirac-type conditions for pancyclicity in $r$-uniform hypergraphs for all $3\leq r\leq n$ when $n$ is sufficiently large. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00130 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pancyclicity in hypergraphs with large uniformity Bailey, Teegan Hollars, Isaiah Li, Yupei Luo, Ruth Combinatorics A Berge cycle of length $\ell$ in a hypergraph $\mathcal{H}$ is a sequence of alternating vertices and edges $v_0e_0v_1e_1...v_\ell e_\ell v_0$ such that $\{v_i,v_{i+1}\}\subseteq e_i$ for all $i$, with indices taken modulo $\ell$. For $n$ sufficiently large and $r\geq \lfloor\frac{n-1}{2}\rfloor-1$ we prove exact minimum degree conditions for an $n$-vertex, $r$-uniform hypergraph to contain Berge cycles of every length between $2$ and $n$. In conjunction with previous work, this provides sharp Dirac-type conditions for pancyclicity in $r$-uniform hypergraphs for all $3\leq r\leq n$ when $n$ is sufficiently large. |
| title | Pancyclicity in hypergraphs with large uniformity |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.00130 |