A graph G is called k-ordered if for every sequence of k distinct vertices there is a cycle traversing these vertices in the given order. In the present paper we consider two novel generalizations of this concept, k-vertex-edge-ordered and strongly k-vertex-edge-ordered. We prove the following results for a chordal graph G: (a) G is (2k-3)-connected if and only if it is k-vertex-edge-ordered (k ≥ 3). (b) G is (2k-1)-connected if and only if it is strongly k-vertex-edge-ordered...