Criterion for active Hamiltonian paths in complete graphs

Characterize which sequences of vertices form active Hamiltonian paths of the complete graph K_n under the recursive path-generation process defined for active paths.

Background

The paper defines active Hamiltonian paths recursively: paths begin with one of the two cyclic orientations of the Hamiltonian cycle and are extended by a prescribed chord-switching operation. Not every Hamiltonian path starting at c_1 is active; the authors give an explicit counterexample in K_6. They ask for a simple criterion deciding whether a vertex sequence is active in K_n.

References

Is there a simple criterion for a sequence of vertices to be an active path of $K_n$?

— Lollipops, dense cycles and chords  (2502.04726 - Dvořák et al., 7 Feb 2025) in Section 5, “Concluding remarks and open problems”