On $r$-Equitable Coloring of Complete Multipartite Graphs

TL;DR

This paper investigates $r$-equitable colorings of complete multipartite graphs, providing necessary and sufficient conditions, and exact values for the minimum number of colors needed for such colorings.

Contribution

It establishes a complete characterization and exact formulas for $r$-equitable colorings in complete multipartite graphs, advancing understanding of graph coloring constraints.

Findings

Provided necessary and sufficient conditions for $r$-equitable colorings.

Derived exact values of $oldsymbol{ ext{} extit{ ext{chi}}_{r=}(G) ext{}}$ and $oldsymbol{ ext{ extit{ ext{chi}}}^*_{r=}(G) ext{}}$.

Extended the theory of equitable coloring to include a parameter $r$ for broader applications.

Abstract

Let $r \geqslant 0$ and $k \geqslant 1$ 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 $k \geqslant n$ is denoted by $\chi^*_{r=} (G)$. 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 $\chi^*_{r=} (G)$.