Planar Turán number of quasi-double stars
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908451709386752 |
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| author | Liu, Huiqing Xie, Tian Zhao, Qin |
| author_facet | Liu, Huiqing Xie, Tian Zhao, Qin |
| contents | Given a graph H, we call a graph $\textit{H-free}$ if it does not contain H as a subgraph. The planar Turán number of a graph H, denoted by $ex_{\mathcal{P}}(n, H)$, is the maximum number of edges in a planar H-free graph on n vertices. A (h,k)-quasi-double star $W_{h,k}$, obtained from a path $P_3=v_1v_2v_3$ by adding h leaves and k leaves to the vertices $v_1$ and $v_3$, respectively, is a subclass of caterpillars. In this paper, we study $ex_{\mathcal{P}}(n,W_{h,k})$ for all $1\le h\le 2\le k\le 5$, and obtain some tight bounds $ex_{\mathcal{P}}(n,W_{h,k})\leq\frac{3(h+k)}{h+k+2}n$ for $3\le h+k\le 5$ with equality holds if $(h+k+2)\mid n$, and $ex_{\mathcal{P}}(n,W_{1,5})\le \frac{5}{2}n$ with equality holds if $12\mid n$. Also we show that $\frac{9}{4}n\le ex_{\mathcal{P}}(n,W_{2,4})\le \frac{5}{2}n$ and $\frac{5}{2}n\le ex_{\mathcal{P}}(n,W_{2,5})\le \frac{17}{6}n$, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11860 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Planar Turán number of quasi-double stars Liu, Huiqing Xie, Tian Zhao, Qin Combinatorics Given a graph H, we call a graph $\textit{H-free}$ if it does not contain H as a subgraph. The planar Turán number of a graph H, denoted by $ex_{\mathcal{P}}(n, H)$, is the maximum number of edges in a planar H-free graph on n vertices. A (h,k)-quasi-double star $W_{h,k}$, obtained from a path $P_3=v_1v_2v_3$ by adding h leaves and k leaves to the vertices $v_1$ and $v_3$, respectively, is a subclass of caterpillars. In this paper, we study $ex_{\mathcal{P}}(n,W_{h,k})$ for all $1\le h\le 2\le k\le 5$, and obtain some tight bounds $ex_{\mathcal{P}}(n,W_{h,k})\leq\frac{3(h+k)}{h+k+2}n$ for $3\le h+k\le 5$ with equality holds if $(h+k+2)\mid n$, and $ex_{\mathcal{P}}(n,W_{1,5})\le \frac{5}{2}n$ with equality holds if $12\mid n$. Also we show that $\frac{9}{4}n\le ex_{\mathcal{P}}(n,W_{2,4})\le \frac{5}{2}n$ and $\frac{5}{2}n\le ex_{\mathcal{P}}(n,W_{2,5})\le \frac{17}{6}n$, respectively. |
| title | Planar Turán number of quasi-double stars |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.11860 |