Optimal Groomings with Grooming Ratios Six and Seven

Grooming uniform all-to-all traffic in optical (SONET) rings with grooming ratio C requires the determination of a decomposition of the complete graph into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The determination of optimal C-groomings has been considered for 3 ≤ C ≤ 9, and completely solved for 3 ≤ C ≤ 5. For C = 6, it has been shown that the lower bound for the drop cost of an optimal C-grooming can be attained for almost all orders with 5 exceptions and 308 possible exceptions. For C = 7, there are infinitely many unsettled orders; especially the case n ≡ 2 (mod 3) is far from complete. In this paper, we show that the lower bound for the drop cost of a 6-grooming can be attained for almost all orders by reducing the 308 possible exceptions to 3, and that the lower bound for the drop cost of a 7-grooming can be attained for almost all orders with seven exceptions and 16 possible exceptions. Moreover, for the unsettled orders, we give upper bounds for the minimum drop costs.