Smallest 2-edge-connected graphs without a spanning trail

In this paper, we prove that if G is a 2-edge-connected simple graph with |V(G)| ≤ 10, then G has a spanning trail with the exception of two graphs. This result also determines the smallest 2edge-connected graphs without a spanning trail. As applications, we get some results concerning the existence of spanning trails and traceability of line graphs under degree-sum conditions. The latter extends a similar theorem in [10]. Our results also correct a flaw of the proof of a result in [10].