On the k-independence number of graph products

The k-independence number of a graph, αk(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 Gk. Although it is known that αk(G) = α(Gk), this, in general, does not hold for most graph products, and thus the existing bounds for α 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.