On the connected Turán number of Berge paths and Berge cycles
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917430835544064 |
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| author | Zhao, Xiamiao Gerbner, Dániel Zhou, Junpeng |
| author_facet | Zhao, Xiamiao Gerbner, Dániel Zhou, Junpeng |
| contents | Given a graph $F$, a Berge copy of $F$ (Berge-$F$ for short) is a hypergraph obtained by enlarging the edges arbitrarily. Győri, Salia and Zamora determined the maximum number of hyperedges in a connected $r$-uniform hypergraph on $n$ vertices containing no Berge path of length $k-1$ for all $k\geq 2r+14$ and sufficiently large $n$, and asked for the minimum $k_0$ such that this extremal number holds for all $k\geq k_0$. In this paper, we prove that the extremal number holds for all $k\geq 2r+2$ and fails for $k\le 2r+1$, thereby completely resolving the problem posed by Győri, Salia and Zamora. Moreover, we improve the result of Füredi, Kostochka and Luo, who determined the maximum number of hyperedges in a $2$-connected $n$-vertex $r$-uniform hypergraph containing no Berge cycle of length at least $k$ for all $k\geq 4r$ and sufficiently large $n$, by showing that this extremal number holds for all $k\geq 2r+2$ and fails for $k\le 2r+1$.
Our approach reduces Berge-Turán problems to classical extremal graph theory problems, and applies recent work of Ai, Lei, Ning and Shi concerning the feasibility of graph parameters and the Kelmans operation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07642 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the connected Turán number of Berge paths and Berge cycles Zhao, Xiamiao Gerbner, Dániel Zhou, Junpeng Combinatorics Given a graph $F$, a Berge copy of $F$ (Berge-$F$ for short) is a hypergraph obtained by enlarging the edges arbitrarily. Győri, Salia and Zamora determined the maximum number of hyperedges in a connected $r$-uniform hypergraph on $n$ vertices containing no Berge path of length $k-1$ for all $k\geq 2r+14$ and sufficiently large $n$, and asked for the minimum $k_0$ such that this extremal number holds for all $k\geq k_0$. In this paper, we prove that the extremal number holds for all $k\geq 2r+2$ and fails for $k\le 2r+1$, thereby completely resolving the problem posed by Győri, Salia and Zamora. Moreover, we improve the result of Füredi, Kostochka and Luo, who determined the maximum number of hyperedges in a $2$-connected $n$-vertex $r$-uniform hypergraph containing no Berge cycle of length at least $k$ for all $k\geq 4r$ and sufficiently large $n$, by showing that this extremal number holds for all $k\geq 2r+2$ and fails for $k\le 2r+1$. Our approach reduces Berge-Turán problems to classical extremal graph theory problems, and applies recent work of Ai, Lei, Ning and Shi concerning the feasibility of graph parameters and the Kelmans operation. |
| title | On the connected Turán number of Berge paths and Berge cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.07642 |