Product Cordial Sets of Long Grids

Abstract

A binary vertex coloring (labeling) \(f: V(G) \to \mathbb{Z}_2\) 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 \(f^*: E(G) \to \mathbb{Z}_2\) defined by \(f^*(uv) = f(u)f(v)\) for all \(uv \in E(G)\). Let \(e_f(i) = |\{uv \in E(G) : f^*(uv) = i\}|\) be the number of edges of \(G\) that are labeled \(i\). The product-cordial index of the labeling \(f\) is the number \(pc(f) = |e_f(0) – e_f(1)|\). The product-cordial set of the graph \(G\), denoted by \(PC(G)\), is defined by

\[PC(G) = \{pc(f): f \text{ is a friendly labeling of } G\}.\]

In this paper, we will determine the product-cordial sets of long grids \(P_m \times P_n\), introduce a class of fully product-cordial trees and suggest new research directions in this topic.