Counting $k$-cycles in $5$-connected planar triangulations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913978579419136 |
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| author | Agrahari, Gyaneshwar Liu, Xiaonan Wang, Zhiyu |
| author_facet | Agrahari, Gyaneshwar Liu, Xiaonan Wang, Zhiyu |
| contents | We show that every $n$-vertex $5$-connected planar triangulation has at most $9n-50$ many cycles of length $5$ for all $n\ge 20$ and this upper bound is tight. We also show that for every $k\geq 6$, there exists some constant $C(k)$ such that for sufficiently large $n$, every $n$-vertex $5$-connected planar graph has at most $C(k) \cdot n^{\lfloor{k/3}\rfloor}$ many cycles of length $k$. This upper bound is asymptotically tight for all $k\geq 6$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_18090 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting $k$-cycles in $5$-connected planar triangulations Agrahari, Gyaneshwar Liu, Xiaonan Wang, Zhiyu Combinatorics 05C10, 05C30, 05C38, 05C40 We show that every $n$-vertex $5$-connected planar triangulation has at most $9n-50$ many cycles of length $5$ for all $n\ge 20$ and this upper bound is tight. We also show that for every $k\geq 6$, there exists some constant $C(k)$ such that for sufficiently large $n$, every $n$-vertex $5$-connected planar graph has at most $C(k) \cdot n^{\lfloor{k/3}\rfloor}$ many cycles of length $k$. This upper bound is asymptotically tight for all $k\geq 6$. |
| title | Counting $k$-cycles in $5$-connected planar triangulations |
| topic | Combinatorics 05C10, 05C30, 05C38, 05C40 |
| url | https://arxiv.org/abs/2507.18090 |