Characterize active Hamiltonian paths in complete graphs

Characterize which sequences of vertices form active Hamiltonian paths of the complete graph K_n under the recursive active-path construction defined for a lollipop cycle.

Background

Active paths are Hamiltonian paths of the lollipop cycle generated recursively from the two initial cyclic orders by chord-based transformations. The paper notes that not every Hamiltonian path starting at c_1 is active, giving an example in K_6, and asks for a simple criterion determining activity in K_n.

References

This fact leads to the two following questions: 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 4, “Concluding remarks and open problems”