Uniform treatment of the exceptional-family path deliverables

Develop a uniform treatment of the Hamiltonian-path deliverables for the two exceptional families of non-stutter graphs—binary signatures with an odd multiplicity and permutations of (2k,1,1)—and thereby provide a uniform path-side formulation corresponding to Lehmer’s original path conjecture.

Background

The paper proves Hamiltonian cycles for the non-stutter graph N(S) outside two exceptional families: binary signatures with an odd multiplicity and permutations of (2k,1,1). For the exceptional families, the paper uses previously known results establishing Hamiltonian paths but no Hamiltonian cycles, primarily to show that the exception set is exact.

The authors note that the original Lehmer conjecture concerns path-like traversals rather than only Hamiltonian cycles of the non-stutter graph. Although the relevant exceptional families are covered individually in the literature, the paper does not provide a single uniform treatment of their path constructions or of the associated path deliverables.

References

The path side. The reformulated conjecture also has a path form at the two exceptional families, and Lehmer's original conjecture is about paths; the present article consumes the path clauses only for exactness. A uniform treatment of the path deliverables remains open, though the literature covers the families.

— A proof of Lehmer's permutation conjecture for neighbor-swap graphs  (2610.01240 - Verhoeff, 1 Oct 2026) in Section Conclusion, subsection “Future work”