On the balanced decomposition number

A balanced coloring of a graph G means a triple {P1, P2,X} of mutually disjoint subsets of the vertex-set V (G) such that V (G) = P1⊎P2⊎X and |P1| = |P2|. A balanced decomposition associated with the balanced coloring V (G) = P1⊎P2⊎X of G is defined as a partition of V (G) = V1⊎· · ·⊎Vr (for some r) such that, for every i ∈ {1, · · · , r}, the subgraphG[Vi] of G is connected and |Vi∩P1| = |Vi∩P2|. Then the balanced decomposition number of a graph G is defined as the minimum integer s such that, for every balanced coloring V (G) = P1⊎P2⊎X of G, there exists a balanced decomposition V (G) = V1 ⊎ · · · ⊎ Vr whose every element Vi(i = 1, · · · , r) has at most s vertices. S. Fujita and H. Liu [SIAM J. Discrete Math. 24, (2010), pp. 1597–1616] proved a nice theorem which states that the balanced decomposition number of a graph G is at most 3 if and only if G is ⌊ |V (G)| 2 ⌋-connected. Unfortunately, their proof is lengthy (about 10 pages) and complicated. Here we give an immediate proof of the theorem. This proof makes clear a relationship between balanced decomposition number and graph matching. keywords: graph decomposition, coloring, connectivity, bipartite matching Email: [email protected] 1