The Irregularity Strength of \(m \times n\) Grids for \(m,n \geq 18\)

Given a graph \(G\) with weighting \(w : E(G) \to \mathbb{Z}^+\), the strength of \(G(w)\) is the maximum weight on any edge. The weight 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 weights are distinct. The irregularity strength of \(G\) is the minimum strength of the graph under all irregular weightings. We determine the irregularity strength of the \(m \times n\) grid for all \(m, n \geq 18\).