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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2210.01190 |
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| _version_ | 1866912425837592576 |
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| author | Lo, On-Hei Solomon Zamfirescu, Carol T. |
| author_facet | Lo, On-Hei Solomon Zamfirescu, Carol T. |
| contents | We investigate the minimum number of cycles of specified lengths in planar $n$-vertex triangulations $G$. It is proven that this number is $Ω(n)$ for any cycle length at most $3 + \max \{ {\rm rad}(G^*), \lceil (\frac{n-3}{2})^{\log_32} \rceil \}$, where ${\rm rad}(G^*)$ denotes the radius of the triangulation's dual, which is at least logarithmic but can be linear in the order of the triangulation. We also show that there exist planar hamiltonian $n$-vertex triangulations containing $O(n)$ many $k$-cycles for any $k \in \{ \lceil n - \sqrt[5]{n} \rceil, \ldots, n \}$. Furthermore, we prove that planar 4-connected $n$-vertex triangulations contain $Ω(n)$ many $k$-cycles for every $k \in \{ 3, \ldots, n \}$, and that, under certain additional conditions, they contain $Ω(n^2)$ $k$-cycles for many values of $k$, including $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_01190 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Counting cycles in planar triangulations Lo, On-Hei Solomon Zamfirescu, Carol T. Combinatorics We investigate the minimum number of cycles of specified lengths in planar $n$-vertex triangulations $G$. It is proven that this number is $Ω(n)$ for any cycle length at most $3 + \max \{ {\rm rad}(G^*), \lceil (\frac{n-3}{2})^{\log_32} \rceil \}$, where ${\rm rad}(G^*)$ denotes the radius of the triangulation's dual, which is at least logarithmic but can be linear in the order of the triangulation. We also show that there exist planar hamiltonian $n$-vertex triangulations containing $O(n)$ many $k$-cycles for any $k \in \{ \lceil n - \sqrt[5]{n} \rceil, \ldots, n \}$. Furthermore, we prove that planar 4-connected $n$-vertex triangulations contain $Ω(n)$ many $k$-cycles for every $k \in \{ 3, \ldots, n \}$, and that, under certain additional conditions, they contain $Ω(n^2)$ $k$-cycles for many values of $k$, including $n$. |
| title | Counting cycles in planar triangulations |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2210.01190 |