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Sep 20, 2021 at 0:47 comment added Louis D One bit of information is that this question is almost a decade old and was posed by a different user. However, I offer a bounty on it because I had a very similar question and wanted to start by "reviving" this old question before asking my own version of it. I am aware of the literature including the paper of Aharoni and Haxell; however, for our purposes it is crucial that we allow for f(G_i)=f(G_j). I have never seen this variant studied in any way, but it seems natural enough that there's some hope.
Sep 19, 2021 at 7:53 history answered domotorp CC BY-SA 4.0