I am thinking about a problem for quite a while now and I can't seem to find some way to tackle it or to find relevant info regarding known results. The problem I'm interested in is as follows. Suppose we have a graph $G(V,E)$ and a function $f$, which maps every edge to another edge. For example edge 1 gets mapped onto edge 3, edge 2 onto edge 4, ..., it can be any kind of mapping. Now we consider an edge coloring on the graph $C_1$. We now swap the colors of the edges according to the function $f$ to get a new coloring $C_2$. In the example edge 3 gets the color of edge 1, edge 4 gets the color of edge 2 and so on. The question I'm now interested in is:
Does there exist a coloring $C_1$ that contains a perfect matching on the graph $G$, such that after the swapping, the coloring $C_2$ also contains a perfect matching on the graph $G$?
What properties does $f$ need to have to satisfy this property?
I know this is a very general question and I have no clue how to tackle it. I do not know if this question is even interesting. Any sort of help or insight is appreciated even if it's a only very specific case.
I'm actually mostly interested in the case of bipartite graphs and 2-color colorings, but any other variation is surely welcome.