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 elf in E(G), G has a spanning (e', elf)-trail. For an integer r ~ 0, a graph is called r-Eulerian-connected if for any X s::: E(G) with IXI ~r, and for any u, v E V(G), G has a spanning (u, v)-trail T such that X s::: E(T). The r-edge-Eulerian­ connectivity of a graph can be defined similarly. Let O(r) be the minimum value of k such that every k-edge-connected graph is r-Eulerian-connected. Catlin proved that 0(0) =4. We shall show that O(r) =4 for 0 ~ r ~ 2, and O(r) = r + 1 for r ~ 3. Results on r-edge-Eulerian connectivity are also discussed. © 2005 Elsevier B.V. All rights reserved.