Donovan Young on Nostr: To see where this two-parameter freedom comes from in the 4-partite case, consider ...

To see where this two-parameter freedom comes from in the 4-partite case, consider the following "moves" that can be made on a given perfect matching:

We could take one edge from the \(p_1\), and another from the \(p_3\). (We have chosen these two because they are non-incident, i.e. the groups \(p_2,p_4,p_5,p_6\) are joined to the \(p_1\) through a vertex. We've also assumed \(p_1,p_3>0\)).

Now we imagine breaking the two chosen edges (somewhere in their middles) and rejoining them in another way: we can either produce an edge to add to \(p_4\) and one to add to \(p_2\), or, rejoining them differently, an edge to add to \(p_5\) and an edge to add to \(p_6\).

This is a two-parameter freedom which can be applied in general to any two non-incident edges in the perfect matching.