On k-edge-ordered graphs Original Research Article

A graph is k-ordered if, for any sequence of k vertices, there is a cycle containing these vertices in the given order. A graph is k-edge-ordered if, for any sequence of k edges, there is a tour containing these edges in the given order. Finally, a graph is strongly k-edge-ordered if for any sequence of k oriented edges, there is a tour containing these edges in the given order and in the given orientations. In this paper, we prove that every 2k-ordered (resp. (2k+1)-ordered) graph is k-edge-ordered (resp. strongly k-edge-ordered). We also examine degree conditions and connectivity for k-edge-ordered graphs, and state results on k-edge-ordered Eulerian graphs.