Total Edge Irregularity Strength of a Categorical Product of Two Paths

Abstract

An edge irregular total \(k\)-labeling of a graph \(G = (V, E)\) is a labeling \(f: V \cup E \to \{1, 2, \ldots, k\}\) such that the total edge-weights \(wt(xy) = f(x) + f(xy) + f(y)\) are distinct for all pairs of distinct edges. The minimum \(k\) for which \(G\) has an edge irregular total \(k\)-labeling is called the total edge irregularity strength of \(G\). In this paper, we determine the exact value of the total edge irregularity strength of the Cartesian product of two paths \(P_n\) and \(P_m\). Our result provides further evidence supporting a recent conjecture of Ivančo and Jendrol.