A Strengthening of a Theorem of Tutte on Hamiltonicity of Polyhedra

Gunnar Brinkmann, Carol T. Zamfirescu

Published 2016-06-06Version 1

In 1956, Tutte showed that every planar 4-connected graph is hamiltonian. In this article, we will prove that even if we relax the prerequisites on the polyhedra from 4-connected to containing at most three 3-cuts, hamiltonicity is still guaranteed. This also strengthens a theorem of Jackson and Yu, who have shown the 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 $k \ge 6$ 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.