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arXiv · 2509.06131

Planar Tur\'{a}n numbers of three configurations

Xuqing Bai·Zhipeng Gao·Ping Li
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Abstract

The planar Tu\'{a}n 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, $\Theta$-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 Tur\'{a}n 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 $\Theta_4$. We further consider the planar Tur\'{a}n numbers of the all graphs obtained by combining $C_3$ and $\Theta_4$. 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.

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Xuqing Bai, Zhipeng Gao, Ping Li. 2025-09-07. Planar Tur\'{a}n numbers of three configurations. https://arxiv.org/abs/2509.06131

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