The per-degree fractional perfect-matching conjecture for r-graphs

Let GG be an rr-graph. A (t,r)(t,r)-PM is a multiset of trt\cdot r perfect matchings of GG such that every edge of GG is contained in exactly tt of them. The per-degree fractional perfect-matching conjecture. For every r1r\geq 1 there is a tr1t_r\geq 1 such that every rr-graph has a (tr,r)(t_r,r)-PM. This is a weaker version of the generalized Berge–Fulkerson conjecture, and the source states that the relevant reductions remain open.

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Primary source

Yulai Ma, Eckhard Steffen, Isaak H. Wolf and Junxue Zhang, “Some conjectures on r-graphs and equivalences”, arXiv:2411.01753 (2026).

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