On the hamiltonicity of a planar graph and its vertex-deleted subgraphs

Tutte proved that every planar 4-connected graph is hamiltonian. Thomassen showed that the same conclusion holds for the superclass of planar graphs with minimum degree at least 4 in which all vertex-deleted subgraphs are hamiltonian. We here prove that if in a planar -vertex graph with minimum degree at least 4 at least vertex-deleted subgraphs are hamiltonian, then the graph contains two hamiltonian cycles, but that for every there exists a nonhamiltonian polyhedral -vertex graph with minimum degree at least 4 containing hamiltonian vertex-deleted subgraphs. Furthermore, we study the hamiltonicity of planar triangulations and their vertex-deleted subgraphs as well as Bondy's meta-conjecture, and prove that a polyhedral graph with minimum degree at least 4 in which all vertex-deleted subgraphs are traceable, must itself be traceable.