Component membership of perfect matchings in arbitrary nanotube resonance graphs

Determine whether two perfect matchings of an arbitrary matchable nanotube lie in the same connected component of its resonance graph.

Background

The paper establishes a criterion for connectedness of the resonance graph of a matchable nanotube: the graph is connected precisely when every nice cycle is an I-cycle or bounds a hexagonal system. It also proves a flow-based equivalence for elementary nanotubes.

For general, non-elementary nanotubes, the authors show that equality of flows across a cut segment is necessary but not sufficient for two perfect matchings to belong to the same component, using a (3,0)-type nanotube whose distinct perfect matchings have equal flow but are isolated. Thus, the component-membership problem for arbitrary nanotubes remains explicitly unresolved in the paper.

References

So we now wonder whether two perfect matchings of a nanotube $N$ lie in the same component of its resonance graph.

— The resonance graphs of coronoid systems and nanotubes  (2608.16687 - Liang et al., 17 Aug 2026) in Section 4.2, immediately after Theorem 4.3 (labeled Theorem \ref{N-connected})