On the Independent Coloring of Graphs with Applications to the Independence Number of Cartesian Product Graphs

Abstract

Let $G$ be a graph with $V=V\left(G\right)$ . A nonempty subset $S$ of $V$ is called an independent set of $G$ if no two distinct vertices in $S$ are adjacent. The union of a class { $S:S$ is an independent set of $G$ } and $\left\{\varnothing \right\}$ is denoted by $\mathcal{I}\left(G\right)$ . For a graph $H$ , a function $f:V⟶\mathcal{I}\left(H\right)$ is called an $H-$ independent coloring of $G$ (or simply called an $H-$ coloring) if $f\left(x\right)\cap f\left(y\right)=\varnothing$ for any adjacent vertices $x,y\in V$ and $f\left(V\right)$ is a class of disjoint sets. Let $\alpha \left(H,G\right)$ denote the maximum cardinality of the set{ ${\sum }_{x\in V}\left|f\left(x\right)\right|:f$ is an $H-$ coloring of $G$ }. In this paper, we obtain basic properties of an $H-$ coloring of $G$ and find $\alpha \left(H,G\right)$ of some families of graphs $G$ and $H$ . Furthermore, we apply them to determine the independence number of the Cartesian product of a complete graph $K_n$ and a graph $G$ and prove that $\alpha \left(K_n\square G\right)=\alpha \left(K_n,G\right)$ .