完全图关联族的精确二阶Zarankiewicz数

中文摘要 AI 辅助

设 $n\ge6$，$m=\binom n2$，并考虑 $K_n$ 上的完全图关联族：列为顶点，行为边，单边图即为关联图。我们研究该族的二阶、带符号及递归线Zarankiewicz数 $z_2,z_{SL},z_{RL}$。Löfberg和Qi的通用胞腔界给出 $$ z_2(m,n)\le Z(n):=\left\lfloor\frac{n(n-1)(n+2)}4\right\rfloor, $$ 其中当 $n$ 为偶数或 $n\equiv1\pmod4$ 时 $Z(n)=n(n-1)(n+2)/4$，当 $n\equiv3\pmod4$ 时 $Z(n)=(n(n-1)(n+2)-2)/4$。我们证明该界可达，即 $z_2=z_{SL}=z_{RL}=Z(n)$，对于偶数 $n=2q$（其中 $q$ 为奇素数）以及奇数 $n=2p+1$（其中 $p$ 为奇数）。构造是嵌套的完美（或近完美）一因子分解；$p$ 的奇偶性控制网格是否带洞饱和。网格簿记精确化：当 $n$ 为偶数或 $n\equiv1\pmod4$ 时 $H=0$，当 $n\equiv3\pmod4$ 时 $H=1$；特别地，$(n(n-1)(n+2)-1)/4$ 永远不会给出该值。我们还通过显式构型解决了另外四个阶数，并记录了其 $(\mathrm{RW}3^+)$ 证书闭包：$n=8,12$（偶数，$q$ 为偶数）和 $n=9,13$（奇数，$p$ 为偶数），分别给出 $140,462,198,585$。$n=13$ 和 $n=12$ 的构型是通过删除一个顶点的星形并重新配对孤立胞腔，从已认证的 $n=14$ 构型获得的。其余阶数（最小为 $n=16$）作为猜想陈述。

英文摘要

For $n\ge6$, $m=\binom n2$, let the complete-graph incidence family on $K_n$ have the vertices of $K_n$ as columns, its edges as rows, and the incidence graph as one-edge graph. The universal cell bound of Löfberg and Qi gives $z_2(m,n)\le Z(n):=\lfloor n(n-1)(n+2)/4\rfloor$. We implement the nested one-factorization construction of that family and determine exactly what it certifies. For $n=2q$ with $q$ an odd prime, $q\ge5$, the construction has no hole and the cross-factor transfer equations apply verbatim, giving $z_2=z_{SL}=z_{RL}=Z(n)$; $n=6$ is settled by a separate cyclic witness. For odd $n=2p+1$ the near-perfect one-factorization yields Hamilton paths closed into odd cycles, and the transfer argument breaks: that scheme certifies only $R(G_p)\le Z(n)$. The orders $n=7,8,9,12,13,16,17,18,20,21$ are settled by explicit configurations of a different shape, each attaining the cell bound and satisfying $(\mathrm{RW}3^+)$, obtained from a larger configuration by deleting vertex stars and repairing the restricted grid. At each of these orders $z_2=z_{SL}=z_{RL}=Z(n)$; all remaining orders are conjectural. The first settled order $n=7$ is the only one whose grid has a hole, grounded by a zero-companion rule. Machine-readable configurations and a certificate checker accompany the paper.

发表机构

• South China Normal University（华南师范大学）
• Linköping University（林雪平大学）
• Jiangsu Provincial Scientific Research Center of Applied Mathematics（江苏省应用数学省级科研中心）
• The Hong Kong Polytechnic University（香港理工大学）

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