AI 中文总结
研究禁止较短奇圈的平面图中奇圈最大数量问题，通过确定\(\mathrm{ex}_\mathcal{P}(n, C_{2k+1}, \{C_3,C_5,\ldots,C_{2k-1}\})\)精确值，证实了关于奇圈的相关猜想。
AI 中文摘要
给定一个图\(H\)和一族图\(\mathcal{F}\)，广义平面Turán数\(\mathrm{ex}_\mathcal{P}(n, H, \mathcal{F})\)是不含\(\mathcal{F}\)中任何子图的\(n\)顶点平面图中\(H\)的最大拷贝数。当仅计算\(H\)的诱导拷贝时，用\(\mathrm{ex}_\mathcal{P}(n, H^{\mathrm{ind}}, \mathcal{F})\)表示相应的广义平面Turán数。Győri和Karim确定了\(\mathrm{ex}_\mathcal{P}(n, C_{5}, \{C_3\})\)。本文确定了对于每个\(k \ge 3\)时\(\mathrm{ex}_\mathcal{P}(n, C_{2k+1}, \{C_3,C_5,\ldots,C_{2k-1}\})\)的精确值。由于禁止了所有较短奇圈，每个\(C_{2k+1}\)都是诱导的。此问题与平面图中奇圈的可诱导性密切相关。Ghosh等人（以及独立的Savery）确定了\(\mathrm{ex}_\mathcal{P}(n, C_5^{\mathrm{ind}}, \emptyset)\)的精确值，并对所有\(k \ge 3\)的奇圈\(C_{2k+1}\)提出了一个猜想。我们的结果在禁止所有较短奇圈的额外假设下证实了他们的猜想。
英文摘要
Given a graph $H$ and a family of graphs $\mathcal{F}$, the generalized planar Turán number $\mathrm{ex}_\mathcal{P}(n, H, \mathcal{F})$ is the maximum number of copies of $H$ in an $n$-vertex planar graph that contains no graph $F \in \mathcal{F}$ as a subgraph. When only induced copies of $H$ are counted, we denote the corresponding generalized planar Turán number by $\mathrm{ex}_\mathcal{P}(n, H^{\mathrm{ind}}, \mathcal{F})$. Győri and Karim determined $\mathrm{ex}_\mathcal{P}(n, C_{5}, \{C_3\})$. In this paper, we determine the exact value of $\mathrm{ex}_\mathcal{P}(n, C_{2k+1}, \{C_3,C_5,\ldots,C_{2k-1}\})$ for every $k \ge 3$. Since all shorter odd cycles are forbidden, every $C_{2k+1}$ is induced. This problem is closely related to the inducibility of odd cycles in planar graphs. Ghosh, Győri, Janzer, Paulos, Salia and Zamora~(and independently Savery) determined the exact value of $\mathrm{ex}_\mathcal{P}(n, C_5^{\mathrm{ind}}, \emptyset)$. Moreover, they established a conjecture for all odd cycles $C_{2k+1}$ with $k \ge 3$. Our result confirms their conjecture under the additional assumption that all shorter odd cycles are forbidden.
