Paired-domination of Cartesian products of graphs and rainbow domination

The most famous open problem involving domination in graphs is Vizing's conjecture which states the domination number of the Cartesian product of any two graphs is at least as large as the product of their domination numbers. We investigate a similar problem for paired-domination, and obtain a lower bound in terms of product of domination number of one factor and 3-packing of the other factor. Some results are obtained by applying a new graph invariant called rainbow domination.