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arXiv 2103.12255math.CO
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关于有限射影平面中的$k$ -簇的个数

On the number of $k$-gons in finite projective planes

Vladislav Taranchuk

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摘要：

设$Π$是$n$阶射影平面，$Γ_Π$是它的Levi图（点线关联图）.对于固定的$k \geq 3$，令$c_{2k}(Γ_Π)$表示$Γ_Π$中的$2k$ -循环数。在本文中，我们表明，$$ c_{2k}(Γ_Π) = \frac{1}{2k}n^{2k} + O(n^{2k-2}), \hspace{0.5cm} n \rightarrow \infty. $$，我们还说明了一个猜想，关于第三和第四大项的渐近数的$2k$ -圈在$Γ_Π$。这一结果也是由Voropaev在2012年独立获得的。令$\text{ex}(v, C_{2k}, \mathcal{C}_{\text{odd}}\cup \{C_4\})$表示在所有长围至少为6的$v$阶二部图中最大的$2k$ -圈数。作为上述结果的推论，我们得到了$$ \text{ex}(v, C_{2k}, \mathcal{C}_{\text{odd}}\cup \{C_4\}) = \left(\frac{1}{2^{k+1}k}-o(1)\right)v^k, \hspace{0.5cm} v \rightarrow \infty. $$

英文摘要：

Let $Π$ be a projective plane of order $n$ and $Γ_Π$ be its Levi graph (the point-line incidence graph). For fixed $k \geq 3$, let $c_{2k}(Γ_Π)$ denote the number of $2k$-cycles in $Γ_Π$. In this paper we show that $$ c_{2k}(Γ_Π) = \frac{1}{2k}n^{2k} + O(n^{2k-2}), \hspace{0.5cm} n \rightarrow \infty. $$ We also state a conjecture regarding the third and fourth largest terms in the asymptotic of the number of $2k$-cycles in $Γ_Π$. This result was also obtained independently by Voropaev in 2012. Let $\text{ex}(v, C_{2k}, \mathcal{C}_{\text{odd}}\cup \{C_4\})$ denote the greatest number of $2k$-cycles amongst all bipartite graphs of order $v$ and girth at least 6. As a corollary of the result above, we obtain $$ \text{ex}(v, C_{2k}, \mathcal{C}_{\text{odd}}\cup \{C_4\}) = \left(\frac{1}{2^{k+1}k}-o(1)\right)v^k, \hspace{0.5cm} v \rightarrow \infty. $$

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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