Let G be a 2-edge-connected simple graph on n>;95 vertices. Let l be the number of vertices of degree 2 in G. We prove that if l<n⧸5−19 and if, for every edge uvϵE(G),d(v)+d(v)⩾2n⧸5−2, then exactly one of the following holds: (a) G has a spanning closed trail; (b) G can be contracted to K2, c−2, where c⩽max{5, 3+l} is an odd number.

An example shows that if a graph satisfies the conditions above except that it has too many vertices of degree 2, then the conclusion fails. This result is related to a conjecture of Benhocine et al. (1986), recently proved by Veldman. We obtain some other related results.