An explicit construction of two completely independent spanning trees in the four-dimensional dual-cube

2026-08-01

AI Summary

This study resolves a long-standing open problem concerning the existence of two completely independent spanning trees in the four-dimensional folded cube $F_4$. By explicitly constructing a cubic polynomial with ten terms over the finite field $\mathbb{F}_2$, the authors define internal vertices via its level sets and employ a solver-free program to verify connectivity, thereby successfully building two completely independent spanning trees using 254 edges. This result completes the characterization that $F_n$ admits two completely independent spanning trees if and only if $n \geq 4$. Moreover, it establishes the optimality of the construction by proving that neither affine nor quadratic polynomials suffice within this algebraic framework, and that ten terms constitute the minimal number required for any cubic rule.

Abstract

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.

Problem

• dual-cube
• completely independent spanning trees
• graph theory
• network reliability
• combinatorial construction

Innovation

• completely independent spanning trees
• dual-cube
• explicit construction
• polynomial method over 𝔽₂
• machine-verified connectivity