Orienting perfect-matching edges in 6-cycle 4-covers

Determine how to orient the perfect-matching edges associated with 6-cycle 4-covers in a natural and useful way, and establish a construction that extends beyond covers splitting into two 6-cycle double covers to general 6-cycle 4-covers.

Background

The paper explains that perfect-matching edges arise naturally in the decomposition underlying a 6-cycle 4-cover, but the authors do not obtain a satisfactory orientation scheme for them. A partial idea works when the cover splits into two 6-cycle double covers and the relevant perfect-matching edges connect the two parts, while its extension to general covers remains unresolved.

References

We could also try to orient the perfect matching itself, even though we haven't succeeded in this endeavour, to come up with something natural looking.

— Computational Graph Decompositions I: Oriented Berge-Fulkerson Conjecture  (2501.05348 - Ulyanov, 9 Jan 2025) in Section “Orienting the perfect matching”