Upper bounds on minimum balanced bipartitions

A balanced bipartition of a graph G is a partition of V ( G ) into two subsets V 1 and V 2 , which differ in size by at most 1 . The minimum balanced bipartition problem asks for a balanced bipartition V 1 , V 2 of a graph minimizing e ( V 1 , V 2 ) , where e ( V 1 , V 2 ) is the number of edges joining V 1 and V 2 . We present a tight upper bound on the minimum of e ( V 1 , V 2 ) , giving one answer to a question of Bollobás and Scott. We prove that every connected triangle-free plane graph G of order at least 3 has a balanced bipartition V 1 , V 2 with e ( V 1 , V 2 ) ≤ | V ( G ) | − 2 , and we show that K 1 , 3 , K 3 , 3 − e , and K 2 , n , with n ≥ 1 , are precisely the extremal graphs. We also show that every plane graph G without separating triangles has a balanced bipartition V 1 , V 2 such that e ( V 1 , V 2 ) ≤ | V ( G ) | + 1 .