The Balanced Decomposition Number and Vertex Connectivity

• Authors: Shinya Fujita;Henry Liu
• Venue: SIAM Journal on Discrete Mathematics
• Year: 2010

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 coloring of a graph $G$ is a coloring of some of the vertices of $G$ with two colors, such that there is the same number of vertices in each color. Then, $f(G)$ is the minimum integer $s$ with the following property: For any balanced coloring of $G$, there is a partition $V(G)=V_1\,\dot\cup\,\cdots\,\dot\cup\,V_r$ such that, for every $i$, $V_i$ induces a connected subgraph, $|V_i|\leq s$, and $V_i$ contains the same number of colored vertices in each color. 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 characterization for noncomplete graphs $G$ of order $n$ which are $\lfloor\frac{n}{2}\rfloor$-connected, in view of the balanced decomposition number. We shall prove that a necessary and sufficient condition for such $\lfloor\frac{n}{2}\rfloor$-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 generalized $\Theta$-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 by Fujita and Liu.