四维对偶立方中两棵完全独立生成树的显式构造
An explicit construction of two completely independent spanning trees in the four-dimensional dual-cube
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中文总结 AI 辅助
该研究解决了四维对偶立方F₄是否存在两棵完全独立生成树的问题，通过显式构造完成了Fₙ存在该树的n≥4的分类，还得出了相关规则的不可行性结果。
中文摘要 AI 辅助
Lalou、Mbarek、Skender和Togni（arXiv:2607.25917）证明，对所有n≥5，n维对偶立方Fₙ都存在两棵完全独立生成树，发现n≤3时不存在，确定F₄是首个未解决的情况，并报告了超过700小时无结果的计算。我们通过显式构造解决了该情况，完成分类：当且仅当n≥4时，Fₙ存在两棵完全独立生成树。两棵树的内部顶点集是F₂上一个含10项的三次多项式在7个顶点比特上的水平集，正确性可简化为有限连通性检查，该检查由随证书发布的无求解器程序进行机器验证。在F₄中，两棵树必然使用256条边中的254条。我们还报告了相同形式更简单规则的精确不可行性结果：在搜索模型中，无仿射或二次规则可行，且三次规则最少需要10项。
英文摘要
Lalou, Mbarek, Skender and Togni (arXiv:2607.25917) proved that the $n$-dimensional dual-cube $F_n$ admits two completely independent spanning trees for every $n\ge 5$, observed that none exist for $n\le 3$, and identified $F_4$ as the first unresolved case, reporting more than 700 hours of inconclusive computation. We settle this case affirmatively by an explicit construction, completing the classification: $F_n$ admits two completely independent spanning trees if and only if $n\ge 4$. The internal-vertex sets of the two trees are the level sets of a single ten-term cubic polynomial over $\mathbb{F}_2$ in the seven vertex bits, and correctness reduces to finite connectivity checks that are machine-verified by a solver-free program distributed with the certificate. In $F_4$ the two trees necessarily use 254 of the 256 edges. We also report exact infeasibility results for simpler rules of the same shape: within the search model, no affine or quadratic rule works, and ten terms is the fewest possible for a cubic rule.
