Chin, Ding, and Wang [CDW05] have shown that examples exist for which the pulling tetrahedralization of a convex polytope is not Hamiltonian. (A pulling tetrahedralization is obtained by joining one vertex (the apex) to all other vertices of the polytope.) It is open if the shelling tetrahedralization may be always Hamiltonian.
Escalona et al. [EFMU07] prove the conjecture up to n = 20: every points set of n≤20 points admits a Hamiltonian Tetrahedralization. They also detail an algorithm that finds a Hamiltonian Tetrahedralization for n points by adding O(n) Steiner points. The algorithm runs in O(n3/2) time.