Every 5-connected planar triangulation is 4-ordered Hamiltonian

A graph $G$ is said to be \textit{$4$-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 \textit{$4$-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.