Bollobás and Scott (Random Struct. Alg. 21 (2002) 414–430) asked for conditions that guarantee a bisection of a graph with m edges in which each class has at most (1/4+o(1))m edges. We demonstrate that cycles of length 4 play an important role for this question. Let G be a graph with m edges, minimum degree δ, and containing no cycle of length 4. We show that if (i) G is 2-connected, or (ii) δ ⩾ 3, or (iii) δ ⩾ 2 and the girth of G is at least 5, then G admits a bisection in which each class has at most (1/4+o(1))m edges. We show that each of these conditions are best possible. On the other hand, a construction by Alon, Bollobás, Krivelevich and Sudakov shows that for infinitely many m there exists a graph with m edges and girth at least 5 for which any bisection has at least (1/4−o(1))m edges in one of the two classes.

Let G be a graph with m edges, and let k be a positive integer. We show that V(G) admits a k-partition V1, . . . Vk such that for i ∈ {1, 2, . . . k}, and , where e(Vi) denotes the number of edges with both ends in Vi and . This answers a problem of Bollobás and Scott [2] in the affirmative. Moreover, for i ∈ {1, 2, . . ., k}, which is close to being best possible and settles another problem of Bollobás and Scott [2].