Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V ( G ) , G has a spanning ( u , v ) -trail. A graph G is edge-Eulerian-connected if for any e ′ and e ″ in E ( G ) , G has a spanning ( e ′ , e ″ ) -trail. For an integer r ⩾ 0 , a graph is called r -Eulerian-connected if for any X ⊆ E ( G ) with | X | ⩽ r , and for any u , v ∈ V ( G ) , G has a spanning ( u , v ) -trail T such that X ⊆ E ( T ) . The r -edge-Eulerian-connectivity of a graph can be defined similarly. Let θ ( r ) be the minimum value of k such that every k -edge-connected graph is r -Eulerian-connected. Catlin proved that θ ( 0 ) = 4 . We shall show that θ ( r ) = 4 for 0 ⩽ r ⩽ 2 , and θ ( r ) = r + 1 for r ⩾ 3 . Results on r -edge-Eulerian connectivity are also discussed