Okay, sure, but the proof that there exists a fractional 3-edge-coloring is either a corollary of Edmond's Theorem or something essentially equivalent to it (such as the approach Seymour uses in his paper on r-graphs). In short, I agree that we are talking about essentially the same argument.

I believe that your first question is open. It is a well known conjecture (elsewhere on this site) that there should be 2 perfect matchings whose intersection does not contain an odd cut.

The second question has been resolved (using Edmond's theorem) by Kaiser, Kral, and Norine and I will add a link to the paper in the reference section.


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