Graph Partitioning and Traffic Grooming with Bounded Degree Request Graph

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 Munoz and San [12] 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 any request graph with maximum degree at most Delta. 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 Delta can be partitioned into subgraphs with at most C edges and each vertex appears in at most M(C, Delta) subgraphs [12]. The cases where Delta = 2 and Delta = 3,C not equal 4 were solved by Munoz and San [12]. In this article we establish the value of M(C, Delta) for many more cases, leaving open only the case where A >= 5 is odd, Delta (mod 2C) is between 3 and C - 1, C >= 4, and the request graph does not contain a perfect matching. In particular, we answer a conjecture of [12].