3-extendability of 8-regular optimal 1-embedded Klein-bottle graphs

Prove that every 8-regular optimal 1-embedded graph on the Klein bottle is 3-extendable.

Background

The paper proves that an optimal 1-toroidal graph of even order is 3-extendable if and only if it is 8-regular. The authors propose the corresponding assertion for 8-regular optimal 1-embedded graphs on the Klein bottle.

The Klein-bottle case requires additional analysis because nontrivial surface-separating curves, specifically equators, may affect the cut and barrier arguments used for matching extendability on the torus. The conjecture therefore asks whether 8-regularity remains sufficient for 3-extendability in this setting.

References

In the end of the paper, we furhter establish the following conjecture concern with matching extendability of those graphs. Every $8$-regular optimal $1$-embedded graph on the Klein bottle is $3$-extendable.

— Connectivity and matching extendability of optimal $1$-embedded graphs on the torus  (2501.02726 - Koizumi et al., 6 Jan 2025) in Conjecture 2, Section Remarks