A note on balanced bipartitions

Abstract

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 . Bollobás 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 max { e ( V 1 ) , e ( V 2 ) } ≤ m / 3 , where e ( V i ) denotes the number of edges of G with both ends in V i . 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 max { e ( V 1 ) , e ( V 2 ) } ≤ ( 1 + o ( 1 ) ) m / 4 , answering a question of Bollobás and Scott in the affirmative. We also provide a sharp lower bound on max { e ( 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 .