The Balanced Decomposition Number and Vertex Connectivity

The balanced decomposition number f(G) of a graph G was introduced by Fujita and Nakamigawa [Discr. Appl. Math., 156 (2008), pp. 3339-3344]. A balanced colouring of a graph G is a colouring of some of the vertices of G with two colours, such that there is the same number of vertices in each colour. Then, f(G) is the minimum integer s with the following property: For any balanced colouring of G, there is a partition V (G) = V1 ∪̇ · · · ∪̇Vr such that, for every i, Vi induces a connected subgraph, |Vi| ≤ s, and Vi contains the same number of coloured vertices in each colour. Fujita and Nakamigawa studied the function f(G) for many basic families of graphs, and demonstrated some applications. In this paper, we shall continue the study of the function f(G). We give a characterisation for non-complete graphs G of order n which are b 2 c-connected, in view of the balanced decomposition number. We shall prove that a necessary and sufficient condition for such b 2 c-connected graphs G is f(G) = 3. We shall also determine f(G) when G is a complete multipartite graph, and when G is a generalised Θ-graph (i.e., a graph which is a subdivision of a multiple edge). Some applications will also be discussed. Further results about the balanced decomposition number also appear in two subsequent papers of Fujita and Liu.