On the existence of super edge-connected graphs with prescribed degrees

Let GG be a connected graph of order nn, minimum degree δ(G)δ(G), and edge-connectivity κ′(G)κ′(G). The graph GG is maximally edge-connected   if κ′(G)=δ(G)κ′(G)=δ(G) and super edge-connected if every minimum edge-cut consists of edges incident with a vertex of minimum degree.A list (d1,…,dn)(d1,…,dn) is graphic   if there is a graph with vertices v1,…,vnv1,…,vn such that d(vi)=did(vi)=di for 1≤i≤n1≤i≤n. A graphic list DD is super edge-connected if DD is the degree list of some super edge-connected graph. We prove that a graphic list DD with least element 1 is super edge-connected if and only if (1) ∑i=1ndi≥2n or (2) ∑i=1ndi=2(n−1) and max{di:1≤i≤n}=n−1max{di:1≤i≤n}=n−1. We also give a necessary and sufficient condition for a graphic list with least entry 2 to be super edge-connected, and we show that every graphic list with least element at least 3 is super edge-connected.