The $k$-independence number of direct products of graphs and Hedetniemi’s conjecture

Abstract

The $k$-independence number of $G$, denoted as ${\alpha }_k\left(G\right)$, is the size of a largest $k$-colorable subgraph of $G$. The direct product of graphs $G$ and $H$, denoted as $G\times H$, is the graph with vertex set $V\left(G\right)\times V\left(H\right)$, where two vertices $(x_1,y_1)$ and $(x_2,y_2)$ are adjacent in $G\times H$, if $x_1$ is adjacent to $x_2$ in $G$ and $y_1$ is adjacent to $y_2$ in $H$. We conjecture that for any graphs $G$ and $H$,

$${\alpha }_k(G\times H)\le {\alpha }_k\left(G\right)\left|V\left(H\right)\right|+{\alpha }_k\left(H\right)\left|V\left(G\right)\right|-{\alpha }_k\left(G\right){\alpha }_k\left(H\right)\text{.}$$

The conjecture is stronger than Hedetniemi’s conjecture. We prove the conjecture for $k=1,2$ and prove that ${\alpha }_k(G\times H)\le {\alpha }_k\left(G\right)\left|V\left(H\right)\right|+{\alpha }_k\left(H\right)\left|V\left(G\right)\right|-{\alpha }_k\left(G\right)\alpha \left(H\right)$ holds for any $k$.