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We address the problem of traffic grooming in WDM rings with all-to-all uniform unitary traffic. We want to minimize the total number of SONET add-drop multiplexers (ADMs) required. This problem corresponds to a partition of the edges of the complete graph into subgraphs, where each subgraph has at most C edges (where C is the grooming ratio) and where the total number of vertices has to be minimized. Using tools of graph and design theory, we optimally solve the problem for practical values and infinite congruence classes of values for a given C. Among others, we give optimal constructions when C N(N 1)=6 and results when C = 12. We also show how to improve lower bounds by using refined counting techniques, and how to use efficiently an ILP program by restricting the search space.