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 ismaximally 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. A list (d1, . . . , dn) is graphic if there is a graph with vertices v1, . . . , vn such that d(vi) = di 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 Dwith least element 1 is super edge-connected if and only if (1) n i=1 di ≥ 2n or (2) n i=1 di = 2(n − 1) and max{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. © 2014 Elsevier B.V. All rights reserved.