Cut and total cut complexes of Möbius ladder graphs

Determine the homotopy types, homology groups, and other topological invariants of the cut complex and total cut complex of the Möbius ladder graphs M_{2n}.

Background

The paper determines the homotopy types of the independence and perfect matching complexes of Möbius ladder graphs M_{2n}, finding periodic behavior according to congruence classes of n. It then identifies the cut complex and total cut complex as other natural simplicial complexes associated with these graphs.

Unlike the complexes treated in the main results, the topology of the cut complex and total cut complex of M_{2n} is not determined in the paper. The authors specifically ask for their homotopy types, homology groups, and other topological invariants, noting that the symmetry of Möbius ladder graphs may lead to analogous periodic behavior.

References

Determining the homotopy types, homology groups, and other topological invariants of the cut complex and the total cut complex of $M_{2n}$ remains an interesting open problem.

— The Homotopy Types of the Independence and Perfect Matching Complex of Möbius Ladder Graph  (2608.30601 - Agarwal et al., 31 Aug 2026) in Section 5, “Future Directions,” first Problem

In particular, it would be interesting to study suitable $4$-regular graphs and determine whether their associated complexes exhibit periodic homotopy types or other systematic topological behavior.

— The Homotopy Types of the Independence and Perfect Matching Complex of Möbius Ladder Graph  (2608.30601 - Agarwal et al., 31 Aug 2026) in Section 5, “Future Directions,” third Problem