拟双星图的尖锐平面Turán界
Sharp planar Turán bounds for quasi-double stars
- Guangzhou University（广州大学）
- Louisiana Christian University（路易斯安那基督教大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究确定拟双星图 $W_{2,4}$、$W_{2,5}$ 和 $W_{3,4}$ 的平面Turán数，给出尖锐边界及精确极值数，填补了已有结果的间隙。
AI中文摘要：
我们研究 $W$-自由平面图，其中 $W\in\{W_{2,4},W_{2,5},W_{3,4}\}$，拟双星图 $W_{h,k}$ 由一条三顶点路径通过在其中一个端点连接 $h$ 个叶子、另一个端点连接 $k$ 个叶子得到。我们证明每个含 $n$ 个顶点的 $W_{2,4}$-自由平面图至多有 $9n/4$ 条边，且当 $8\mid n$ 时该界可达到。这确定了 $W_{2,4}$ 的平面Turán密度为 $9/4$。我们还为 $W_{2,5}$ 建立了尖锐上界 $5n/2$。结合已知的最大度为五的平面图构造，对于每个 $n\ge15$，得到 $\ex_{\PP}(n,W_{2,5})=\lfloor5n/2\rfloor$。这些结果填补了Liu等人界中两个相应的系数间隙。我们的证明使用了高度数顶点的结构限制、局部删除以及半径为二的邻域中的度数亏缺。对于 $W_{3,4}$，我们刻画了具有支配顶点且避免该树的平面图，并确定了它们的精确极值数 $\lfloor(5n-7)/2\rfloor$，对每个 $n\ge10$ 成立。最后，对于 $W=W_{2,4},W_{2,5},W_{3,4}$，$W$-自由平面三角剖分分别至多有八、十二和十一个顶点；前两个界是尖锐的。
英文摘要：
We study $W$-free planar graphs for $W\in\{W_{2,4},W_{2,5},W_{3,4}\}$, where the quasi-double star $W_{h,k}$ is obtained from a three-vertex path by attaching $h$ leaves to one endpoint and $k$ leaves to the other. We prove that every $W_{2,4}$-free planar graph on $n$ vertices has at most $9n/4$ edges, and the bound is attained whenever $8\mid n$. This determines the planar Turán density of $W_{2,4}$ as $9/4$. We also establish the sharp upper bound $5n/2$ for $W_{2,5}$. Combined with known constructions of planar graphs of maximum degree five, it yields $\ex_{\PP}(n,W_{2,5})=\lfloor5n/2\rfloor$ for every $n\ge15$. These results close the two corresponding coefficient gaps in the bounds of Liu et~al. Our proofs use structural restrictions on high-degree vertices, local deletions, and degree deficits in neighborhoods of radius two. For $W_{3,4}$, we characterize the planar graphs with a dominating vertex that avoid this tree and determine their exact extremal number, $\lfloor(5n-7)/2\rfloor$, for every $n\ge10$. Finally, $W$-free planar triangulations have at most eight, twelve, and eleven vertices for $W=W_{2,4},W_{2,5},W_{3,4}$, respectively; the first two bounds are sharp.
