On minimum balanced bipartitions of triangle-free graphs

A balanced bipartition of a graph G is a partition of V ( G ) into two subsets V 1 and V 2 that differ in cardinality by at most 1. A minimum balanced bipartition of G is a balanced bipartition V 1 , V 2 of G minimizing e ( V 1 , V 2 ), where e ( V 1 , V 2 ) is the number of edges joining V 1 and V 2 and is usually referred to as the size of the bipartition. In this paper, we show that every 2-connected graph G admits a balanced bipartition V 1 , V 2 such that the subgraphs of G induced by V 1 and by V 2 are both connected. This yields a good upper bound to the size of minimum balanced bipartition of sparse graphs. We also present two upper bounds to the size of minimum balanced bipartitions of triangle-free graphs which sharpen the corresponding bounds of Fan et al. (Discrete Math. 312:1077–1083, 2012 ).