Equitable colorings of Cartesian products of graphs

Abstract

The present paper studies the following variation of vertex coloring on graphs. A graph $G$ is equitably $k$-colorable if there is a mapping $f:V\left(G\right)\to \{1,2,\dots ,k\}$ such that $f\left(x\right)\ne f\left(y\right)$ for $xy\in E\left(G\right)$ and $\mid \mid f^{-1}\left(i\right)\mid -\mid f^{-1}\left(j\right)\mid \mid \le 1$ for $1\le i,j\le k$. The equitable chromatic number of a graph $G$, denoted by ${\chi }_{=}\left(G\right)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The equitable chromatic threshold of a graph $G$, denoted by ${\chi }_{=}^{\ast }\left(G\right)$, is the minimum $t$ such that $G$ is equitably $k$-colorable for all $k\ge t$. Our focus is on the equitable colorability of Cartesian products of graphs. In particular, we give exact values or upper bounds of ${\chi }_{=}\left(G\square H\right)$ and ${\chi }_{=}^{\ast }\left(G\square H\right)$ when $G$ and $H$ are cycles, paths, stars, or complete bipartite graphs.