**Definition:** Let $MH(n)$ be the maximal number of perfect matchings (1-regular graphs) on $n$ vertices where the union of any two perfect matchings is a Hamiltonian cycle.

**Question:** Is it true that $MH(N)=n-1$ for infinitely many $n$?

**Additional information:** Needless to say that the problem makes sense only when $n$ is even. I needed in a proof that $3 \leq MH(n)$. This is true and not hard, but right now I can prove it by two simple constructions, one when $n=4k$ and an other when $n=4k+2$. So I am just curious what might be the maximal possible number.

It is easy to see that $MH(N)\leq n-1$ since the perfect matchings must be edge disjoint and there are not enough edges for more than $n-1$. It is very easy to see that $MH(4)=3$. But I was quite surprised when it turned out that $MH(6)=5$. See the picture for the construction, it is not hard to check that indeed every pairwise union forms a Hamiltonian path. So right now it is still possible that $MH(n)=n-1$ for all even $n$.