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, u-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.