On $r$-Equitable Coloring of Complete Multipartite Graphs

Let $r⩾0$ and $k⩾1$ be integers. We say that a graph $G$ has an $r$-equitable $k$-coloring if there exists a proper $k$-coloring of $G$ such that the sizes of any two color classes differ by at most $r$. The least $k$ such that a graph $G$ has an $r$-equitable $k$-coloring is denoted by ${\chi }_{r=}(G)$, and the least $n$ such that a graph $G$ has an $r$-equitable $k$-coloring for all $k⩾n$ is denoted by ${\chi }_{r=}^{\ast }(G)$. In this paper, we propose a necessary and sufficient condition for a complete multipartite graph $G$ to have an $r$-equitable $k$-coloring, and also give exact values of ${\chi }_{r=}(G)$ and ${\chi }_{r=}^{\ast }(G)$.