We study the Max k-Cut problem and its dual, the Min k-Partition problem. In the Min k-Partition problem, given a graph G=(V,E) and positive edge weights, we want to find an edge set of minimum weight whose removal makes G k-colorable. For the Max k-Cut problem we show that, if P&neq;NP, no polynomial time approximation algorithm can achieve a relative error better than 1/34k. It is well known that a relative error of 1/k is obtained by a naive randomized heuristic.

For the Min k-Partition problem, we show that for k>2 and for every epsilon>0, there exists a constant alpha such that the problem cannot be approximated within alpha|V^(2-epsilon)|, even for dense graphs. Both problems are directly related to the frequency allocation problem for cellular (mobile) telephones, an application of industrial relevance