Supereulerian graphs and the Petersen graph

A graph G is supereulerian if G has a spanning eulerian subgraph. Boesch et al. [J. Graph Theory, 1, 79–84 (1977)] proposed the problem of characterizing supereulerian graphs. In this paper, we prove that any 3-edge-connected graph with at most 11 edge-cuts of size 3 is supereulerian if and only if it cannot be contractible to the Petersen graph. This extends a former result of Catlin and Lai [J. Combin. Theory, Ser. B, 66, 123–139 (1996)].