On a conjecture on the balanced decomposition number

The concept of balanced decomposition number was introduced by Fujita and Nakamigawa in connection with a simultaneous transfer problem. A balanced colouring of a graph G is a pair ( R , B ) of disjoint subsets R , B ⊆ V ( G ) with | R | = | B | . A balanced decomposition D of a balanced colouring C = ( R , B ) of G is a partition of vertices V ( G ) = V 1 ∪ V 2 ∪ ⋯ ∪ V r such that G [ V i ] is connected and | V i ∩ R | = | V i ∩ B | for 1 ≤ i ≤ r . Let C be the set of all balanced colourings of G and D ( C ) be the set of all balanced decompositions of G for C ∈ C . Then the balanced decomposition number f ( G ) of G is f ( G ) = max C ∈ C min D ∈ D ( C ) max 1 ≤ i ≤ r | V i | . Fujita and Nakamigawa conjectured that if G is a 2-vertex connected graph of n vertices, then f ( G ) ≤ ⌊ n 2 ⌋ + 1 . In this paper, we confirm this conjecture in the affirmative.