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

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