4-connected triangulations and 4-orderedness

For a positive integer k ≥ 4 , a graph G is called k -ordered, if for any ordered set of k distinct vertices of G , G has a cycle that contains all the vertices in the designated order. Goddard (2002) [3] showed that every 4-connected triangulation of the plane is 4-ordered. In this paper, we improve this result; every 4-connected triangulation of any surface is 4-ordered. Our proof is much shorter than the proof by Goddard.