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Femtocell networks have great potential for mobile applications. However, interference due to the co-existence of macrocells and femtocells is a serious problem. In addition, dense femtocells introduce severe inter-system interference. In this paper, a graph theory based dynamic sub-band allocation technique is presented to avoid downlink interference. We model the cells and their mutual interference as graph elements, nodes and weighted edges, respectively. To maintain a tolerable interference level, the total bandwidth is divided into a number of sub-bands, and these are assigned to the femtocells using a graph coloring algorithm. The division is optimized to minimize the bandwidth used to meet user traffic requirements, and minimize the femto-to-macro interference. In addition, sub-bands are assigned to the femtocells to avoid inter-femto interference. An iterative spectrum management algorithm is also introduced.