On Hamiltonian Tetrahedralizations Of Convex Polyhedra

Let Tp denote any tetrahedralization of a convex polyhedron P and let G be the dual graph of Tp such that each node of G T corresponds to a tetrahedron of Tp and two nodes are connected by an edge in G T if and only if the two corresponding tetrahedra share a common facet in Tp. Tp is called a Hamiltonian tetrahedralization if G contains a Hamiltonian path (HP). A wellknown open problem in computational geometry is: can every polytope in 3D be partitioned into tetrahedra such that the dual graph has an HP? In this note, we shall show that there exists a 92-vertex polyhedron in which the pulling method does not yield a Hamiltonian tetrahedralization, here the pulling method is the simplest method to ensure a linear-size decomposition and is one of the most commonly used tetrahedralization methods for convex polyhedra. Furthermore, we can construct a convex polyhedron with n vertices such that the longest path in the dual graph in question can be as short as O(1). This fact suggests that it may not be possible to find a good approximation of a HP for convex polyhedra using the pulling method.