On the Rank Number of Grid Graphs
  A vertex k-ranking is a labeling of the vertices of a graph with integers from 1 to k so any path connecting two vertices with the same label will pass through a vertex with a greater label. The rank number of a graph is defined to be the minimum possible k for which a k-ranking exists for that graph. For mxn grid graphs, the rank number has been found only for m<4. In this paper, we determine its for m=4 and improve its upper bound for general grids. Furthermore, we improve lower bounds on the rank numbers for square and triangle grid graphs from logarithmic to linear. These new lower bounds are key to characterizing the rank number for general grids, and our results have applications in optimizing VLSI circuit design and parallel processing, search, and scheduling. 
Sitan Chen
math.CO