Planar Turán numbers of three configurations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915493323997184 |
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| author | Bai, Xuqing Gao, Zhipeng Li, Ping |
| author_facet | Bai, Xuqing Gao, Zhipeng Li, Ping |
| contents | The planar Tuán number of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is defined as the maximum number of edges in an $n$-vertex $H$-free planar graph. The exact value of $ex_{\mathcal{P}}(n,H)$ remains a mystery when $H$ is large (for example, $H$ is a long path or a long cycle), while tight bounds have been established for many small planar graphs such as cycles, paths, $Θ$-graphs and other small graphs formed by a union of them.
One representative graph among such union graphs is $K_1+L$ where $L$ is a linear forest without isolated vertices.
Previous works solved the cases when $L$ is a path or a matching.
In this work, we first investigate the planar Turán number of the graph $K_1+L$ when $L$ is the disjoint union of a $P_2$ and $P_3$.
Equivalently, $K_1+L$ represents a specific configuration formed by combining a $C_3$ and a $Θ_4$.
We further consider the planar Turán numbers of the all graphs obtained by combining $C_3$ and $Θ_4$. Among the six possible such configurations, three have been resolved in earlier works. For the remaining three configurations (including
$K_1+(P_2\dot{\cup}P_3)$), we derive tight bounds. Furthermore, we completely characterize all extremal graphs for the remaining two of these three cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06131 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Planar Turán numbers of three configurations Bai, Xuqing Gao, Zhipeng Li, Ping Combinatorics The planar Tuán number of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is defined as the maximum number of edges in an $n$-vertex $H$-free planar graph. The exact value of $ex_{\mathcal{P}}(n,H)$ remains a mystery when $H$ is large (for example, $H$ is a long path or a long cycle), while tight bounds have been established for many small planar graphs such as cycles, paths, $Θ$-graphs and other small graphs formed by a union of them. One representative graph among such union graphs is $K_1+L$ where $L$ is a linear forest without isolated vertices. Previous works solved the cases when $L$ is a path or a matching. In this work, we first investigate the planar Turán number of the graph $K_1+L$ when $L$ is the disjoint union of a $P_2$ and $P_3$. Equivalently, $K_1+L$ represents a specific configuration formed by combining a $C_3$ and a $Θ_4$. We further consider the planar Turán numbers of the all graphs obtained by combining $C_3$ and $Θ_4$. Among the six possible such configurations, three have been resolved in earlier works. For the remaining three configurations (including $K_1+(P_2\dot{\cup}P_3)$), we derive tight bounds. Furthermore, we completely characterize all extremal graphs for the remaining two of these three cases. |
| title | Planar Turán numbers of three configurations |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.06131 |