Are there any special properties known about the set of perfect matchings of $K_{n,n}$? Like any global structure of this set? Some natural way to partition it? Like is there some algebraic structure on this set? Some group that is known to may be act nicely on this set? (this might be a broad question, so feel free to just link to something you know that goes this way!)
Given a perfect matching of $K_{n,n}$ is there a systematic way to generate other perfect matchings (disjoint) from it?
How large a set of mutually disjoint perfect matchings of $K_{n,n}$ can be obtained ? And how to obtain such a set?
- That count of $\sum_{i=0}^{d-1} (n-i)!$ is what I think is the number of ways one can pick $d$ mutually disjoint perfect matchings of $K_{n,n}$ (the $k^{th}$ summand above corresponds to the number of ways the $k^{th}$ perfect matching can be chosen having made the previous choices)
- $d-1 = n$ is the largest $d$ for which the summand stops being defined. So naively I feel $d = n +1$ is the largest $d$ for which one can find $d$ mutually disjoint perfect matchings. Though I don't know if this d is the largest possible. (by cyclic permutations I seem to be able to generate atmost only $n$ mutually disjoint perfect matchings)
I saw some related papers like, http://www.tau.ac.il/~nogaa/PDFS/mincg3.pdf and http://math.cmu.edu/~af1p/Texfiles/matchplusbip.pdf
A related MO discussion, Postnikov's approach to perfect matchings of graphs
Feel free to may be generalize any of the above questions to the case of more general bipartite graphs or general graphs!