Given a matching M in the hypercube Qⁿ, the profile of M is the vector x=(x₁,…, xₙ) ∈ Nⁿ such that M contains xᵢ edges whose endpoints differ in the ith coordinate. If M is a perfect matching, then it is clear that ||x||₁ = 2ⁿ⁻¹ and it is easy to show that each xᵢ must be even. Verifying a special case of a conjecture of Balister, Gy{{o}}ri, and Schelp, we show that these conditions are also sufficient.
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Joshua Erde (2024) studied this question.
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