ON THE IRREGULARITY STRENGTH OF THE M X N GRID

Given a graph G with weighting w: E(G) --> Z+, the strength of G(w) is the maximum weight on any edge. The sum of a vertex in G(w) is the sum of the weights of all its incident edges. The network G(w) is irregular if the vertex sums are distinct. The irregularity strength of G is the minimum strength of the graph under all irregular weightings. In this paper we determine the irregularity strength of the m x n grid for certain m and n. In particular, for every positive integer d we find the irregularity strength for all but a finite number of m x n grids where n - m = d. In addition, we present a general lower bound for the irregularity strength of graphs.