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 Bollobas 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.

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