The strong equitable vertex 1-arboricity of complete bipartite graphs and balanced complete k-partite graphs

An \emph{equitable $(q, r)$-tree-coloring} of a graph $G$ is a $q$-coloring of $G$ such that the subgraph induced by each color class is a forest of maximum degree at most $r$ and the sizes of any two color classes differ by at most $1.$ Let the \emph{strong equitable vertex $r$-arboricity} of a graph $G,$ denoted by $va^\equiv_r (G)$, be the minimum $p$ such that $G$ has an equitable $(q, r)$-tree-coloring for every $q\geq p.$ The values of $va^\equiv_1 (K_{n,n})$ were investigated by Tao and Lin and Wu, Zhang, and Li where exact values of $va^\equiv_1 (K_{n,n})$ were found in some special cases. In this paper, we extend their results by giving the exact values of $va^\equiv_1 (K_{n,n})$ 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 $va^\equiv_1 (K_{m,n})$ and $va^\equiv_1 (G)$ where $G$ is a balanced complete $k$-partite graph $K_{n,\ldots,n}.$