Product Cordial Sets of Long Grids

A binary vertex coloring (labeling) f : V (G) → Z2 of a graph G is said to be friendly if the number of vertices labeled 0 is almost the same as the number of vertices labeled 1. This friendly labeling induces an edge labeling f∗ : E(G) → Z2 defined by f∗(uv) = f(u)f(v) for all uv ∈ E(G). Let ef (i) = |{uv ∈ E(G) : f∗(uv) = i}| be the number of edges of G that are labeled i. Product-cordial index of the labeling f is the number pc(f) = |ef (0) − ef (1)|. The product-cordial set of the graph G, denoted by PC(G), is defined by PC(G) = {pc(f) : f is a friendly labeling of G }. In this paper, we will determine the product-cordial sets of long grids Pm×Pn, introduce a class of fully product-cordial trees and suggest new research directions in this topic.