A bipartition of the vertex set of a graph is called balanced if the sizes of the sets in the bipartition differ by at most one. Bollobás and Scott [3] conjectured that if G is a graph with minimum degree at least 2 then V (G) admits a balanced bipartition V 1 , V 2 such that for each i, G has at most |E(G)|/3 edges with both ends in V i . The minimum degree condition is necessary, and a result of Bollobás and Scott [5] shows that this conjecture holds for regular graphs G (i.e., when ∆(G) = δ(G)). We prove this conjecture for graphs G with ∆(G) ≤ 7 5 δ(G); hence it holds for graphs G with δ(G) ≥ 5 7 |V (G)|.