Generalized Turán problem for directed cycles
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916764197060608 |
|---|---|
| author | Grzesik, Andrzej Jaworska, Justyna Kielak, Bartłomiej Kuc, Piotr Ślusarczyk, Tomasz |
| author_facet | Grzesik, Andrzej Jaworska, Justyna Kielak, Bartłomiej Kuc, Piotr Ślusarczyk, Tomasz |
| contents | For integers $k, \ell \geq 3$, let $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ denote the maximum number of directed cycles of length $k$ in any oriented graph on $n$ vertices which does not contain a directed cycle of length $\ell$. We establish the order of magnitude of $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ for every $k$ and $\ell$ and determine its value up to a lower error term when $k \nmid \ell$ and $\ell$ is large enough. Additionally, we calculate the value of $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ for some other specific pairs $(k, \ell)$ showing that a diverse class of extremal constructions can appear for small values of $\ell$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_22189 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Turán problem for directed cycles Grzesik, Andrzej Jaworska, Justyna Kielak, Bartłomiej Kuc, Piotr Ślusarczyk, Tomasz Combinatorics For integers $k, \ell \geq 3$, let $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ denote the maximum number of directed cycles of length $k$ in any oriented graph on $n$ vertices which does not contain a directed cycle of length $\ell$. We establish the order of magnitude of $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ for every $k$ and $\ell$ and determine its value up to a lower error term when $k \nmid \ell$ and $\ell$ is large enough. Additionally, we calculate the value of $\mathrm{ex}(n, \overrightarrow{C_k}, \overrightarrow{C_\ell})$ for some other specific pairs $(k, \ell)$ showing that a diverse class of extremal constructions can appear for small values of $\ell$. |
| title | Generalized Turán problem for directed cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.22189 |