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 vertical bar X vertical bar < 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 X-0 subset of E(G) with vertical bar X-0 vertical bar <= 3 and any distinct edges e, e' is an element of E(G), G has a spanning (e, e')-trail containing all edges in X-0, 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]. (C) 2013 Elsevier B.V. All rights reserved.