A 3-semi-perfect 1-factorization of the six-dimensional hypercube
Abstract
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.
Disclosure
“t https://github.com/GLambard/q6-semi-perfect-factorization. Funding No specific funding was received for performing the present work. Declaration of competing interests There are no competing interests to be declared. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this work, the author used OpenAI Terra/Sol 5.6 (High) models to assist computational exploration, independent-code drafting, literature organizat”
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