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An equitable k-coloring of a graph G is a proper k-coloring of G such that the sizes of any two color classes differ by at most one. An equitable (q,r)-tree-coloring of a graph G is an equitable q-coloring of G such that the subgraph induced by each color class is a forest of maximum degree at most r. Let the strong equitable vertex r-arboricity of a graph G, denoted by var≡(G), be the minimum p such that G has an equitable (q,r)-tree-coloring for every q≥p. The values of va1≡(Kn,n) were investigated by Tao and Lin and Wu, Zhang, and Li where exact values of va1≡(Kn,n) were found in some special cases. In this paper, we extend their results by giving the exact values of va1≡(Kn,n) for all cases. In the process, we introduce a new function related to an equitable coloring and obtain a more general result by determining the exact value of each va1≡(Km,n) and va1≡(G) where G is a balanced complete k-partite graph Kn,…,n. Both complete bipartite graphs Km,n and balanced complete k-partite graphs Kn,…,n are symmetry in several aspects and also studied broadly. For the other aspect of symmetry, by the definition of equitable k-coloring of graphs, in a specific case that the number of colors divides the number of vertices of graph, we can say that the graph is a balanced k-partite graph.

The Strong Equitable Vertex 1-Arboricity of Complete Bipartite Graphs and Balanced Complete k-Partite Graphs