Rich-edge parity in perfect matchings

Prove that, in every oriented 6-cycle 4-cover whose vertices are all disordered, each perfect matching contains an even number of rich edges.

Background

The paper reports several experimentally observed properties of oriented and unoriented 6-cycle 4-covers. A rich edge is an edge that occurs in four distinct local edge-pair combinations in the cover, while a disordered vertex is one of the two local vertex-orientation types defined earlier. The stated parity property was verified computationally for small snarks but remains unproved.

References

If we have an o6c4c solution with all disordered vertices, then in each perfect matching we have an even number of rich edges.

— Computational Graph Decompositions I: Oriented Berge-Fulkerson Conjecture  (2501.05348 - Ulyanov, 9 Jan 2025) in Section “More conjectures”