On the Cordial of Weak Labeling of an l-fold Cycle Graph, T_l(C_n)

We investigate a number-theoretic graph labeling known as k-prime cordial labeling, where each vertex of a graph G is assigned a label from the set {1,2, ... , k}, and each edge receives a label equal to the greatest common divisor of its endpoint labels. A weak labeling is k-prime cordial if the number of vertices labeled with each integer differs by at most two, and the number of edges labeled 1 differs from those not labeled 1 by at most two. In this paper, we introduce the concept of the $\ell$-fold of a graph, denoted T_l(G), constructed by joining corresponding vertices across l copies of a base graph G. We focus on the case where G is a cycle graph C_n and show that T_l(C_n) admits a 4-prime cordial weak labeling for all l≥2. This result extends previous work on trigraphs and contributes to the broader understanding of cordial labeling in replicated graph structures.