Hamiltonicity of the face lattice of every polytope

Prove that the cover graph of the face lattice of every polytope is Hamiltonian, thereby establishing the conjecture that every polytope has a Hamiltonian face-lattice cover graph.

Background

The paper studies Hamilton cycles in the cover graphs of face lattices. It proves the conjectured property for all simplicial polytopes and, by duality, for all simple polytopes, substantially extending previously known special cases.

The remaining conjecture concerns arbitrary polytopes, including those that are neither simplicial nor simple. The paper’s proof relies on partitioning the face lattice into cubes arising from simplicial facets, and the conclusion notes that extending the result to all polytopes appears difficult using this approach.