Every 5-connected planar triangulation is 4-ordered Hamiltonian

A graph G is said to be 4-ordered if for any ordered set of four distinct vertices of G, there exists a cycle in G that contains all of the four vertices in the designated order. Furthermore, if we can find such a cycle as a Hamiltonian cycle, G is said to be 4-ordered Hamiltonian. It was shown that every 4-connected planar triangulation is (i) Hamiltonian (by Whitney) and (ii) 4-ordered (by Goddard). Therefore, it is natural to ask whether every 4-connected planar triangulation is 4-ordered Hamiltonian. In this paper, we give a partial solution to the problem, by showing that every 5- connected planar triangulation is 4-ordered Hamiltonian.