Balanced decomposition of a vertex-colored graph

Abstract

A balanced vertex-coloring of a graph G is a function c from V ( G ) to { − 1 , 0 , 1 } such that ∑ { c ( v ) : v ∈ V ( G ) } = 0 . A subset U of V ( G ) is called a balanced set if U induces a connected subgraph and ∑ { c ( v ) : v ∈ U } = 0 . A decomposition V ( G ) = V 1 ∪ ⋯ ∪ V r is called a balanced decomposition if V i is a balanced set for 1 ≤ i ≤ r . In this paper, the balanced decomposition number f ( G ) of G is introduced; f ( G ) is the smallest integer s such that for any balanced vertex-coloring c of G , there exists a balanced decomposition V ( G ) = V 1 ∪ ⋯ ∪ V r with | V i | ≤ s for 1 ≤ i ≤ r . Balanced decomposition numbers of some basic families of graphs such as complete graphs, trees, complete bipartite graphs, cycles, 2-connected graphs are studied.