Closed formula for the positive matching decomposition of hypercubes

Determine a closed formula for pmd(Qn), the positive matching decomposition number of the n-dimensional hypercube.

Background

The paper disproves the preceding tree-product conjecture when four or more factors are K2 by exhibiting a discrepancy for Q4. This also gives a negative answer to an earlier conjecture concerning hypercubes. Although upper bounds are obtained for Qn, the exact dependence of pmd(Qn) on n remains unresolved.

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