On the Existence of Cn-Supermagic Labelings in Cycle-Based Graph Structures

In this paper, we prove the existence of Cn-supermagic labeling for the graph G(n, n) formed by connecting a vertex of n copies of the Cn (n-edge cycle) graph to a vertex of a main Cn graph. Labeling functions enable the edges and the vertices of the graph to be labeled with subgraphs that resemble the structure of Cn. Cn-supermagic labeling requires the sum of the labels to be constant for each Cn-cover, and the labels must be uniquely distributed across the graph. The study examines different labelings of G(n, n) based on the value of n modulo 4, demonstrating their validity through examples. The results show that such labelings are possible and can be applied consistently for specific values of n.