AI 中文总结
研究六维超立方体的 1 - 因子分解，给出使\(G[F]\cong K_{3,3}\)的明确构造，解决了此前的例外情况，构造以有限证书形式呈现，正确性可通过论文表格或两个验证器检查。
AI 中文摘要
对于超立方体\(Q_d\)的一个 1 - 因子分解\(F = \{M_1,\ldots,M_d\}\)，令\(G[F]\)的顶点集为\(F\)，当\(M_i\cup M_j\)是哈密顿圈时\(M_iM_j\)为一条边。Behague 证明了除可能\(k = \ell = 3\)外，对于所有正整数\(k,\ell\)，\(Q_{k + \ell}\)有一个 1 - 因子分解\(F\)使得\(G[F]\cong K_{k,\ell}\)。我们给出了\(Q_6\)的一个明确的 1 - 因子分解，使得\(G[F]\cong K_{3,3}\)，解决了这个例外情况。该构造以有限证书形式给出，其正确性可直接从论文中的表格或随证书提供的两个独立的简短标准库验证器之一进行检查。
英文摘要
For a 1-factorization $F=\{M_1,\ldots,M_d\}$ of the hypercube $Q_d$, let $G[F]$ have vertex set $F$, with $M_iM_j$ an edge exactly when $M_i\cup M_j$ is a Hamilton cycle. Behague proved that $Q_{k+\ell}$ has a 1-factorization $F$ with $G[F]\cong K_{k,\ell}$ for all positive $k,\ell$ except possibly $k=\ell=3$. We give an explicit 1-factorization of $Q_6$ for which $G[F]\cong K_{3,3}$, resolving the exceptional case. The construction is supplied as a finite certificate. Its correctness can be checked directly from the tables in the paper or by either of two independent, short, standard-library verifiers supplied with the certificate.
Comments5 pages. The certificate, two independent Python verifiers, hashes, reproduction instructions, and complete search provenance are archived at https://doi.org/10.5281/zenodo.21404470. Source repository: https://github.com/GLambard/q6-semi-perfect-factorization
