Degree sum of a pair of independent edges and Z 3-connectivity

Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv a E(G), define d(e) = d(u) + d(v). Let F denote the set of all simple 2-edge-connected graphs on n aeyen 4 vertices such that G a F if and only if d(e) + d(e') aeyen 2n for every pair of independent edges e, e' of G. We prove in this paper that for each G a F, G is not Z (3)-connected if and only if G is one of K (2,n-2), K (3,n-3), K (2,n-2) (+) , K (3,n-3) (+) or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 2010, 310: 3390-3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308: 6233-6240].