On the balanced decomposition number

TL;DR

This paper provides a concise proof of a theorem relating the balanced decomposition number of a graph to its connectivity, clarifying the connection with graph matching and simplifying previous lengthy proofs.

Contribution

The paper offers an immediate, clearer proof of the theorem linking balanced decomposition number and graph connectivity, improving understanding of their relationship.

Findings

The balanced decomposition number is at most 3 if and only if the graph is sufficiently connected.

The new proof simplifies the previous complex proof from 2010.

The proof reveals a relationship between balanced decomposition number and graph matching.

Abstract

A {\em balanced coloring} of a graph $G$ means a triple $\{P_1,P_2,X\}$ of mutually disjoint subsets of the vertex-set $V(G)$ such that $V(G)=P_1 \uplus P_2 \uplus X$ and $|P_1|=|P_2|$. A {\em balanced decomposition} associated with the balanced coloring $V(G)=P_1 \uplus P_2 \uplus X$ of $G$ is defined as a partition of $V(G)=V_1 \uplus \cdots \uplus V_r$ (for some $r$) such that, for every $i \in \{1,\cdots,r\}$, the subgraph $G[V_i]$ of $G$ is connected and $|V_i \cap P_1| = |V_i \cap P_2|$. Then the {\em balanced decomposition number} of a graph $G$ is defined as the minimum integer $s$ such that, for every balanced coloring $V(G)=P_1 \uplus P_2 \uplus X$ of $G$, there exists a balanced decomposition $V(G)=V_1 \uplus \cdots \uplus V_r$ whose every element $V_i (i=1, \cdots, r)$ 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 $\lfloor\frac{|V(G)|}{2}\rfloor$-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.