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We derive lower bounds on the bandwidth requirement for feasible channel assignment in a hexagonal cellular network for multimedia communications. An understanding of such bounds would greatly enhance the ability of service providers to offer multimedia based applications over cellular networks. For simplicity, we consider only two types of multimedia signals, with single demand for each type at each node in a 2-band buffering environment, where interference does not extend beyond two cells. Our approach considers the feasibility of assignment at each node, based on all previously assigned channels to its neighbors. The derived lower bounds are tighter than the one reported in the literature, with an improvement of the bounds by about 17% under typical situations.