Balanced kk-decompositions of graphs

For a given integer k≥2k≥2, a balanced  kk-coloring   of a graph GG is a mapping c:V(G)→{0,1,2,…,k}c:V(G)→{0,1,2,…,k} such that |Aj|=|Aj′||Aj|=|Aj′| for 1≤j