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) = V1 ∪ V2 ∪ . . . ∪ Vr such that G[Vi] is connected and |Vi ∩R| = |Vi ∩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 |Vi|. Fujita and Nakamigawa conjectured that If G is a 2-connected graph of n vertices, then f(G) ≤ n2 + 1. The present paper confirms this conjecture.