Graphs with small balanced decomposition numbers

Abstract

A balanced coloring of a graph $G$ is an ordered pair $(R,B)$ of disjoint subsets $R,B \subseteq V(G)$ with $|R|=|B|$. The balanced decomposition number $f(G)$ of a connected graph $G$ is the minimum integer $f$ such that for any balanced coloring $(R,B)$ of $G$ there is a partition $\mathcal{P}$ of $V(G)$ such that $S$ induces a connected subgraph with $|S| \le f$ and $|S \cap R| = |S \cap B|$ for $S \in \mathcal{P}$. This paper gives a short proof for the result by Fujita and Liu (2010) that a graph $G$ of $n$ vertices has $f(G)=3$ if and only if $G$ is $\lfloor \frac{n}{2} \rfloor$-connected but is not a complete graph.