On the $k$-independence number of graph products

Abstract

The $k$-independence number of a graph, $\alpha_k(G)$, is the maximum size of a set of vertices at pairwise distance greater than $k$, or alternatively, the independence number of the $k$-th power graph $G^k$. Although it is known that $\alpha_k(G)=\alpha(G^k)$, this, in general, does not hold for most graph products, and thus the existing bounds for $\alpha$ of graph products cannot be used. In this paper we present sharp upper bounds for the $k$-independence number of several graph products. In particular, we focus on the Cartesian, tensor, strong, and lexicographic products. Some of the bounds previously known in the literature for $k=1$ follow as corollaries of our main results.