On the cordial labeling of \(T_l(P_n)\)

Abstract

Let \(G\) be a simple graph that has \(n\) vertices and \(m\) edges. Let \(V(G)\) be the set of all vertices of \(G\) and \(E(G)\) the set of all edges of \(G\). Let \(f\colon V(G)\to\{1,2,\dots,k\}\) be a function that assigns to each vertex \(v\in V(G)\) a positive integer \(f(v)\in\{1,2,\dots,k\}\). This integer is called the label of \(v\). For each edge \(uv\in E(G)\) we assign a label which is the gcd \((f(u),f(v))\). Let \(v_f(i)\) be the number of vertices labeled with \(i\) for \(i\geq 1\). The function \(f\) is called \(k\)-prime cordial labeling of \(G\) if \(|v_f(i)-v_f(j)|\leq 1\) for all \(i,j\in\{1,2,\dots,k\}\) and \(|e_f(1)-e_f(0)|\leq 1\) where \(e_f(1)\) denotes the number of edges that are labeled with 1 and \(e_f(0)\) denotes the number of edges that are not labeled with 1. In this paper we introduce the concept of the \(l\)-fold of a graph \(G\), \(T_l(G)\), and we show that the \(l\)-fold of a path \(P_n\), \(T_l(P_n)\), is a 4-prime cordial graph.