Berge Pancyclic hypergraphs
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929566721769472 |
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| author | Bailey, Teegan Li, Yupei Luo, Ruth |
| author_facet | Bailey, Teegan Li, Yupei Luo, Ruth |
| contents | A Berge cycle of length $\ell$ in a hypergraph is an alternating sequence of $\ell$ distinct vertices and $\ell$ distinct edges $v_1,e_1,v_2, \ldots, v_\ell, e_{\ell}$ such that $\{v_i, v_{i+1}\} \subseteq e_i$ for all $i$, with indices taken modulo $\ell$. We call an $n$-vertex hypergraph pancyclic if it contains Berge cycles of every length from $3$ to $n$. We prove a sharp Dirac-type result guaranteeing pancyclicity in uniform hypergraphs: for $n \geq 70$, $3 \leq r \leq \lfloor (n-1)/2\rfloor - 2$, if $\cH$ is an $n$-vertex, $r$-uniform hypergraph with minimum degree at least ${\lfloor (n-1)/2 \rfloor \choose r-1} + 1$, then $\cH$ is pancyclic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21733 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Berge Pancyclic hypergraphs Bailey, Teegan Li, Yupei Luo, Ruth Combinatorics A Berge cycle of length $\ell$ in a hypergraph is an alternating sequence of $\ell$ distinct vertices and $\ell$ distinct edges $v_1,e_1,v_2, \ldots, v_\ell, e_{\ell}$ such that $\{v_i, v_{i+1}\} \subseteq e_i$ for all $i$, with indices taken modulo $\ell$. We call an $n$-vertex hypergraph pancyclic if it contains Berge cycles of every length from $3$ to $n$. We prove a sharp Dirac-type result guaranteeing pancyclicity in uniform hypergraphs: for $n \geq 70$, $3 \leq r \leq \lfloor (n-1)/2\rfloor - 2$, if $\cH$ is an $n$-vertex, $r$-uniform hypergraph with minimum degree at least ${\lfloor (n-1)/2 \rfloor \choose r-1} + 1$, then $\cH$ is pancyclic. |
| title | Berge Pancyclic hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.21733 |