## Equitable colorings of Cartesian products of graphs.(English)Zbl 1241.05035

Summary: The present paper studies the following variation of vertex coloring on graphs. A graph $$G$$ is equitably $$k$$-colorable if there is a mapping $$f: V(G)\to\{1,2,\dots,k\}$$ such that $$f(x)\not\in f(y)$$ for $$xy\in E(G)$$ and $$\| f^{-1}(i)|-|f^{-1}(j)\|\leq 1$$ for $$1\leq i,\,j\leq k$$. The equitable chromatic number of a graph $$G$$, denoted by $$\chi_=(G)$$, is the minimum $$k$$ such that $$G$$ is equitably $$k$$-colorable. The equitable chromatic threshold of a graph $$G$$, denoted by $$\chi^*_=(G)$$, is the minimum $$t$$ such that $$G$$ is equitably $$k$$-colorable for all $$k\geq t$$.
Our focus is on the equitable colorability of Cartesian products of graphs. In particular, we give exact values or upper bounds of $$\chi_= (G\square H)$$ and $$\chi^*_=(G\square H)$$ when $$G$$ and $$H$$ are cycles, paths, stars, or complete bipartite graphs.

### MSC:

 05C15 Coloring of graphs and hypergraphs 05C76 Graph operations (line graphs, products, etc.)
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### References:

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