Fall colouring of bipartite graphs and cartesian products of graphs

The question of whether a graph can be partitioned into kk independent dominating sets, which is the same as having a fall  kk-colouring  , is considered. For k=3k=3, it is shown that a graph GG can be partitioned into three independent dominating sets if and only if the cartesian product G□K2G□K2 can be partitioned into three independent dominating sets. The graph K2K2 can be replaced by any graph HH such that there is a mapping f:Qn→Hf:Qn→H, where ff is a type-II graph homomorphism.The cartesian product of two trees is considered, as well as the complexity of partitioning a bipartite graph into three independent dominating sets, which is shown to be NP-complete. For other values of kk, iterated cartesian products are considered, leading to a result that shows for what values of kk the hypercubes can be partitioned into kk independent dominating sets.