Characterization of 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 from a Hamiltonian cycle.

Background

The paper defines active Hamiltonian paths recursively: paths initially following either orientation of the Hamiltonian cycle generate further paths through specified chord exchanges. Not every Hamiltonian path beginning at c_1 is active; the authors give an example in K_6 and explain that six of the 120 such paths are non-active. They explicitly leave open the search for a simple criterion determining activity in K_n.

References

Back to the original definition of active paths, we have several questions. First, is every Hamiltonian path starting in $c_1$ active? The answer is negative, since in the complete graph $K_6$ with the Hamiltonian cycle $(c_1,\dots ,c_6)$, the path $Q=c_1c_2c_5c_4c_3c_6$ is not active (there are 6 non active paths among the 120 paths starting by $c_1$).

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