On 3-edge-connected supereulerian graphs in graph family C(l,k)C(l,k)

Let l>0l>0 and k≥0k≥0 be two integers. Denote by C(l,k)C(l,k) the family of 2-edge-connected graphs such that a graph G∈C(l,k)G∈C(l,k) if and only if for every bond S⊂E(G)S⊂E(G) with |S|≤3|S|≤3, each component of G−SG−S has order at least (|V(G)|−k)/l(|V(G)|−k)/l. In this paper we prove that if a 3-edge-connected graph G∈C(12,1)G∈C(12,1), then GG is supereulerian if and only if GG cannot be contracted to the Petersen graph. Our result extends some results by Chen and by Niu and Xiong. Some applications are also discussed.