Rank numbers of grid graphs

A ranking of a graph is a labeling of the vertices with positive integers such that any path between vertices of the same label contains a vertex of greater label. The rank number of a graph is the smallest possible number of labels in a ranking. We find rank numbers of the Mِbius ladder, K s × P n , and P 3 × P n . We also find bounds for rank numbers of general grid graphs P m × P n .