Call a triangulation of a smooth manifold Hamiltonian if its 1-skeleton has a Hamiltonian cycle. I have several questions about these that I haven't been able to find answers to. First, every smooth manifold has a triangulation. Is it true that every smooth manifold (with boundary) has a Hamiltonian triangulation?
It's also true that any two triangulations of smooth manifolds have a common refinement. Is it true that any two Hamiltonian triangulations have a common refinement which is Hamiltonian? This would be true if the refinement of a Hamiltonian triangulation is a Hamiltonian triangulation, but I'm not sure if it is. Is the barycentric subdivision of a Hamiltonian triangulation Hamiltonian?
If not, are these true in low dimension? In dimension one they are true.