Spanning trails in essentially 4-edge-connected graphs

A connected graph  G  is essentially 4-edge-connected if for any edge cut  X  of  G  with  |X|<4 , either  G−X  is connected or at most one component of  G−X  has edges. In this paper, we introduce a reduction method and investigate the existence of spanning trails in essentially 4-edge-connected graphs. As an application, we prove that if  G  is 4-edge-connected, then for any edge subset  X0⊆E(G)  with  |X0|≤3  and any distinct edges  e,e′∈E(G) ,  G  has a spanning  (e,e′) -trail containing all edges in  X0 , which solves a conjecture posed in [W. Luo, Z.-H. Chen, W.-G. Chen, Spanning trails containing given edges, Discrete Math. 306 (2006) 87–98].