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.