Higher-dimensional Barnette conjecture for simple polytopes

Determine whether every simple d-polytope with d≥4 is hamiltonian.

Background

The paper identifies the higher-dimensional form of Barnette's conjecture, which asserts that every simple polytope of dimension at least four has a hamiltonian graph. The conjecture concerns simple polytopes, whose duals are simplicial, whereas the paper's main constructions produce nontraceable simplicial polytopes. The authors explicitly note that the dual graphs of their constructed polytopes are hamiltonian, so their construction does not provide counterexamples to this conjecture; consequently, the conjecture remains unresolved in the paper.

References

Barnette's conjecture for simple $4$-polytopesConj.~1.5 has the following higher-dimensional form. Every simple $d$-polytope with $d\ge4$ is hamiltonian. The duals of our polytopes are hamiltonian by Proposition~\ref{prop:dual-hamiltonian}, so this construction yields no counterexamples to Conjecture~\ref{conj:Barnette-simple-hamiltonian}.

— Connectivity, toughness, and hamiltonicity of simplicial polytopes  (2610.12836 - Pineda-Villavicencio, 9 Oct 2026) in Section "Existing nonhamiltonicity results for higher-dimensional polytopes" and Conjecture (Barnette's conjecture, higher-dimensional form), near the end of Section 1