Vertex-count congruence for all-rich snark covers

Prove that if a snark has a 6-cycle 4-cover with all edges rich, then its number of vertices is congruent to 2 modulo 4.

Background

The paper identifies snarks whose 6-cycle 4-covers have no poor edges and observes computationally that the number of vertices appears to satisfy a fixed congruence condition. The conjecture formalizes this observation for every snark admitting such an all-rich cover.

References

If we have a 6c4c solution for a snark with all rich edges, then the number of vertices is $4k+2$.

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