A 3-semi-perfect 1-factorization of the six-dimensional hypercube

Abstract

For a 1-factorization $F=\{M_1,\dots ,M_d\}$ of the hypercube $Q_d$, let $G[F]$ have vertex set $F$, with $M_i{M}_j$ 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.

5 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