Cutting a graph into two dissimilar halves

Given a graph G and a subset S of the vertex set of G, the discrepancy of S is defined as the difference between the actual and expected numbers of the edges in the subgraph induced on S . We show that for every graph with n vertices and e edges, n < e < n(n 1)/4, there is an n/2-element subset with the discrepancy of the order of magnitude of Vne . For graphs with fewer than n edges, we calculate the asymptotics for the maximum guaranteed discrepancy of an n/2-element subset . We also introduce a new notion called "bipartite discrepancy" and discuss related results and open problems . 1 . INTRODUCTION Let G be an arbitrary graph with u(G) = n vertices and e (G) = e edges . For any subset S of the vertex set of G, let the discrepancy of S be defined as the Journal of Graph Theory, Vol . 12, No . 1, (1988) 121-131 1) 1988 by John Wiley & Sons, Inc . CCC 0364-9024.88%010121-11$04 .00 122 JOURNAL OF GRAPH THEORY difference between the actual and expected numbers of edges in G[S], i .e ., in the subgraph of G induced by S . That is, let 'll dis (S) = e(S) e J = e(S) e S (lS 1)