Connectivity of 8-regular optimal 1-embedded Klein-bottle graphs

Prove that every 8-regular optimal 1-embedded graph on the Klein bottle has connectivity either 6 or 8.

Background

The paper establishes that every 8-regular optimal 1-toroidal graph has connectivity either 6 or 8 and uses the absence of minimal 7-cuts as a key ingredient. The authors seek an analogous result for optimal 1-embedded graphs on the Klein bottle, where the topology is more complicated because the Klein bottle contains a nontrivial surface-separating simple closed curve, called an equator.

The conjecture specifically asserts that an 8-regular optimal 1-embedded graph on the Klein bottle cannot have connectivity 7. Establishing it would extend the toroidal connectivity result to the Klein bottle and clarify the role of equators in the structure of minimal cuts.

References

We believe that, at least, optimal $1$-embedded graphs on the Klein bottle has no minimal $7$-cut. In other words, we propose the following conjecture, which is similar to the proposition for O1TGs. Every $8$-regular optimal $1$-embedded graph $G$ on the Klein bottle has connectivity either $6$ or $8$.

— Connectivity and matching extendability of optimal $1$-embedded graphs on the torus  (2501.02726 - Koizumi et al., 6 Jan 2025) in Conjecture 1, Section Remarks