Hamiltonian-connectedness of triangulations with few separating triangles

We prove that 3-connected plane triangulations containing a single edge contained in all separating triangles are hamiltonian-connected. As a direct corollary we have that 3-connected plane triangulations with at most one separating triangle are hamiltonian-connected. In order to show bounds on the strongest form of this theorem, we proved that for any s >= 4 there are 3-connected triangulation with s separating triangles that are not hamiltonian-connected. We also present computational results which show that all `small' 3-connected triangulations with at most 3 separating triangles are hamiltonian-connected