Upper and lower bounds of a class of channel assignment problems in cellular networks

A cellular network is often modelled as a graph and the channel assignment problem is formulated as a coloring problem of the graph. We introduce the notion of cellular graphs that models the hexagonal cell structures of a cellular network. Exploiting the regular structure of the cellular graphs we compute the upper and the lower bounds for a class of channel assignment problems. Assuming a k-band buffering system where the interference does not extend beyond k cells away from the call originating cell, we provide two different formulations of the channel assignment problem-distance-k chromatic number problem and k-band chromatic bandwidth problem. We give one algorithm for the first problem and two for the second, with all three algorithms assigning channels to the cells. The complexity of the algorithm for the first problem is O(p), where p is the number of cells. For the second problem, the complexity of the first algorithm is O(p) and the complexity of the second algorithm is O(k⁵log k). All the algorithms are asymptotically optimal, in the sense that the order of the upper bound of the number of channels required is the same as the order of the lower bound