Closed formula for the pmd of hypercubes

Determine a closed formula for pmd(Qn), where Qn is the n-dimensional hypercube.

Background

The paper explains that the tree-product conjecture fails for at least four K2 factors, since the upper bound from Corollary 2.4 is not attained for Q4. This failure also gives a negative answer to an earlier conjecture concerning hypercubes.

Consequently, the exact behavior of pmd(Qn) is unresolved, and the authors identify the existence of a closed formula as an open problem.

References

This gives a negative answer to our previous conjecture (see [4, Conjecture 4.5]) so a possible closed formula for pmd(Qn) remains unknown.

— Positive matching decompositions of the cartesian product of graphs  (2502.02826 - Ghouchan et al., 5 Feb 2025) in Remark following Conjecture 2.5, Section 2, p. 5