Polyhedra with Few 3-Cuts are Hamiltonian

In 1956, Tutte showed that every planar 4-connected graph is hamiltonian. In this article, we will generalize this result and prove that polyhedra with at most three 3-cuts are hamiltonian. In 2002 Jackson and Yu have shown this result for the subclass of triangulations. We also prove that polyhedra with at most four 3-cuts have a hamiltonian path. It is well known that for each k ≥ 6 non-hamiltonian polyhedra with k 3-cuts exist. We give computational results on lower bounds on the order of a possible non-hamiltonian polyhedron for the remaining open cases of polyhedra with four or five 3-cuts.