On non-hamiltonian polyhedra without cubic vertices and their vertex-deleted subgraphs

Thomassen proved in 1978 that if in an n-vertex planar graph G whose minimum degree is at least 4, all vertex-deleted subgraphs of G are hamiltonian, then G is hamiltonian. It was recently shown that in the preceding sentence, all" can be replaced by at least n - 5". In this note we prove that, even if 3-connectedness is assumed, it cannot be replaced by n-24 (or any other integer greater than 24). The exact threshold remains unknown. We show that the same conclusion holds for triangulations and use computational means to prove that, under a natural restriction, this result is best possible.