Channel graphs have long been recognized as a useful tool in studying the blocking probabilities of switching networks. A channel graph is said to be superior to another channel graph if the blocking probability of the former never exceeds that of the latter assuming common but arbitrary link occupancies for the two graphs. Proving superiority has become a standard way to compare channel graphs in the literature. It turns out that many such comparisons can be abstracted as optimal partition problems. In an optimal partition problem, we have a set , and a cost function defined on a partition of . The problem is to find the partition which minimizes . In this paper we study some optimal partition problems when assumes some special forms. We also show that our result has direct applications to the comparison of channel graphs with respect to superiority.