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For a given integer k≥2, a balanced k-coloring of a graph G is a mapping c:V(G)→{0,1,2,…,k} such that |Aj|=|Aj′| for 1≤j<j′≤k, where Aj={v∈V(G):c(v)=j} for 0≤j≤k. The balanced k-decomposition number fk(G) of G is the minimum integer s with the property that for any balanced k-coloring c there is a partition V(G)=V1∪V2∪⋯∪Vr such that Vi induces a connected subgraph with |Vi|≤s and |Vi∩Aj|=|Vi∩Aj′|...
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