Cutting a Graph into Two Dissimilar

S G S e d Given a graph and a subset of the vertex set of , the discrepancy of is defined as th ifference between the actual and expected numbers of the edges in the subgraph induced on. We ¡ S n 2 ¢ £ £ £ £ £) n (n −1 4 s n e n <e < how that for every graph with vertices and edges, , there is an-element subset with the discrepancy of the order of magnitude of. For graphs with fewer than edges we √ ¤ ¥ ¤ ¦ ¤ n ne n 2 o i calculate the asymptotics for the maximum guaranteed discrepancy of an-element subset. We als ntroduce a new notion called the bipartite discrepancy and discuss related results and open problems.