Rich-edge parity for even cycles in split 6-cycle double covers

Prove that, for every 6-cycle 4-cover that splits into two 6-cycle double covers, each possible nowhere-zero 2-flow even cycle contains an even number of rich edges; additionally, when the cover is oriented, prove that the number of circuits and the number of disordered-disordered rich edges are even.

Background

This conjecture concerns 6-cycle 4-covers that decompose into two 6-cycle double covers. It asserts parity constraints simultaneously for the even cycles arising from nowhere-zero 2-flows and, in the oriented case, for the number of circuits and disordered-disordered edges. The paper presents the claim as an experimentally supported statement that it cannot currently prove.

References

If we have a 6c4c solution which splits into two 6-cycle double covers, then each possible nz2- even cycle has an even number of rich edges. Moreover, if it is an o6c4c solution, then we additionally have an even number of circuits, and an even number of drd edges (which is same as saying that the sum of numbers of ordered vertices and rich edges is even).

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