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In this paper, we address the problem of traffic grooming in wavelength-division multiplexing rings with all-to-all and its generalization to many-to-many service by using network coding. We consider minimizing the number of line terminating equipment on two types of unidirectional rings, namely, single-hub and unhubbed rings, as our objective. In single-hub rings, we investigate the minimum cost provisioning of uniform all-to-all traffic in two cases: where network coding is used to linearly combine data, and where it is not used and data are transmitted without coding. We generalize the service mode to many-to-many and evaluate the cost of provisioning. In unhubbed ring, we propose a multihub approach to obtain the minimum cost provisioning in the case of all-to-all and many-to-many traffic. In each type of ring topology, two network scenarios are considered: first, the distinct communication groups in the ring are node-disjoint, and second, the different groups may have common member nodes. From our numerical results, we find that under many-to-many traffic pattern for both scenarios, network coding can reduce the network cost by 10%-20% in single-hub rings and 1%-5% in unhubbed rings in both network scenarios.