Edge-odd graceful labeling of Cartesian product of two paths

Let G be a finite simple graph with p vertices and q edges is called an edge odd graceful labeling if there exists a bijective map f : E(G)→{1, 3, 5,,(2k − 1)} such that when each vertex is assigned, the sum of all edges incident to it mod 2k, where k = max(p, q), the resulting vertex labels are distinct. Any (p, q) - graph admitting edge - odd graceful labeling is called an edge-odd graceful graph. In this paper, we investigate the existence of edge odd graceful labeling for the cartesian product of two paths m P and n P where m and n are any positive integers.