A balanced bipartition of a graph G is a bipartition V 1 and V 2 of V(G) such that -1≤|V 1|-|V 2|≤1. Bollobs and Scott conjectured that if G is a graph with m edges and minimum degree at least 2 then G admits a balanced bipartition V 1,V 2 such that maxe(V 1),e(V 2)≤m3, where e(Vi) denotes the number of edges of G with both ends in Vi. In this note, we prove this conjecture for graphs with average degree at least 6 or with minimum degree at least 5. Moreover, we show that if G is a graph with m edges and n vertices, and if the maximum degree Δ(G)=o(n) or the minimum degree δ(G)→∞, then G admits a balanced bipartition V 1,V 2 such that maxe(V 1),e(V 2) ≤(1+o(1))m4, answering a question of Bollobs and Scott in the affirmative. We also provide a sharp lower bound on maxe(V 1,V 2):V 1,V 2 is a balanced bipartition of G, in terms of size of a maximum matching, where e(V 1,V 2) denotes the number of edges between V 1 and V 2. © 2010 Elsevier B.V.
Xu, B., Yan, J., & Yu, X. (2010). A note on balanced bipartitions. Discrete Mathematics, 310(20), 2613–2617. https://doi.org/10.1016/j.disc.2010.03.029