Minimal Partitions of a Graph

For a given graph \(G\), we fix \(s\), and partition the vertex set into \(s\) classes, so that any given class contains few edges. The result gives a partition \((U_1, U_2, \ldots, U_s)\), where \(e(U_i) \leq \frac{e(G)}{s^2} + 4s\sqrt{e(G)}\) for each \(1 \leq i \leq s\). The error term is compared to previous results for \(s = 2^P\) \({[6]}\), and to a result by Bollobás and Scott \({[1]}\).