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

Let G be a connected graph of order n , minimum degree δ ( G ) , and edge-connectivity κ ′ ( G ) . The graph G is maximally edge-connected if κ ′ ( G ) = δ ( G ) and super edge-connected if every minimum edge-cut consists of edges incident with a vertex of minimum degree. ( d 1 , … , d n ) is graphic if there is a graph with vertices v 1 , … , v n such that d ( v i ) = d i for 1 ≤ i ≤ n . A graphic list D is super edge-connected if D is the degree list of some super edge-connected graph. We prove that a graphic list D with least element 1 is super edge-connected if and only if (1) ∑ i = 1 n d i ≥ 2 n or (2) ∑ i = 1 n d i = 2 ( n − 1 ) and max { d i : 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.