On 3-total edge product cordial labeling of grid

Let us consider a mapping phi:E(G) ->{0, 1, ... ,k - 1} of a graph G, where k is an integer, 2 <= k <=|E(G)|. The mapping phi induces for every vertex v of G the label phi*(v) =Pi (uv is an element of E)(G)(phi(uv))(modk). Let e(phi)(i) (v(phi)(i)) denote the number of edges (vertices) in G that are labeled with the number i under the labeling phi, 0 <= i <= k - 1. The function phi is called a k-total edge product cordial labeling of G if |(e(phi)(i) + v phi(i)) - (e(phi)(j) + v(phi)(j))|<= 1 for 0 <= i < j <= k - 1. A graph G with a k-total edge product cordial labeling is called a k-total edge product cordial graph. In this paper, we prove that the grid graph P-m square P-n for m,n >= 2 admits a 3-total edge product cordial labeling.