We address the problem of estimating the spectrum required in a wireless network for a given demand and interference pattern. This problem can be abstracted as a generalization of the graph coloring problem, which typically presents additional degree of hardness compared to the standard coloring problem. It is worthwhile to note that the question of estimating the spectrum requirement differs markedly from that of allocating channels. The main focus of this work is to obtain strong upper and lower bounds on the spectrum requirement, as opposed to the study of spectrum allocation/management. While the relation to graph coloring establishes the intractability of the spectrum estimation problem for arbitrary network topologies, useful bounds and algorithms are obtainable for specific topologies. We establish some new results regarding generalized coloring, which we use to derive tight bounds for specific families of graphs. We also examine the hexagonal grid topology, a commonly used topology for wireless networks. We design efficient algorithms that exploit the geometric structure of the hexagonal grid topology to determine upper bounds on the spectrum requirement for arbitrary demand patterns. The slack in our upper bounds is estimated by analyzing subgraphs with specific properties. While we consider the worst-case demand patterns to evaluate the performance of our algorithms, we expect them to perform much better in practice.
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