Degree sequences of optimally edge-connected multigraphs

Let u, v be distinct vertices of a multigraph G with degrees d(u) and d(v), respectively. The number of edge-disjoint u, v-paths in G is bounded above by min{d(u), d(v)}. A multigraph G is optimally edge-connected if for all pairs of distinct vertices u and v this upper bound is achieved. If G is a multigraph with degree sequence D, then we say G is a realisation of D. We characterize degree sequences of multigraphs that have an optimally edge-connected realisation as well as those for which every realisation is optimally edge-connected.