**Abstract** : We study a graph partitioning problem which arises from traffic grooming in optical networks. We wish to minimize the equipment cost in a SONET WDM ring network by minimizing the number of Add-Drop Multiplexers (ADMs) used. We consider the version introduced by Mu~noz and Sau~[Mu~noz and Sau, WG 08] where the ring is unidirectional with a grooming factor $C$, and we must design the network (namely, place the ADMs at the nodes) so that it can support \\emphany request graph with maximum degree at most Δ. This problem is essentially equivalent to finding the least integer $M(C,\\Delta)$ such that the edges of any graph with maximum degree at most Δ can be partitioned into subgraphs with at most $C$ edges and each vertex appears in at most $M(C,\\Delta)$ subgraphs~[Mu~noz and Sau, WG 08] . The cases where $\\Delta=2$ and $\\Delta=3,C\\neq 4$ were solved by Mu~noz and Sau~[Mu~noz and Sau, WG 08] . In this article we establish the value of $M(C,\\Delta)$ for many more cases, leaving open only the case where $\\Delta \\geq 5$ is odd, $\\Delta \\pmod2C$ is between $3$ and $C-1$, $C\\geq 4$, and the request graph does not contain a perfect matching. In particular, we answer a conjecture of~[Mu~noz and Sau, WG 08] .