A Study of Fibonacci Cordial Labeling in Structured Graph Families

Abstract: A \emph{Fibonacci cordial labeling} of a graph ( G ) is an injective function ( f: V(G) \rightarrow {F_0, F_1, \dots, F_n} ), where ( F_i ) denotes the ( i^{\text{th}} ) Fibonacci number, such that the induced edge labeling ( f^*: E(G) \rightarrow {0,1} ), given by ( f^*(uv) = (f(u) + f(v)) ) $(\bmod\ 2)$, satisfies the balance condition ( |e_f(0) - e_f(1)| \le 1 ). Here, ( e_f(0) ) and ( e_f(1) ) represent the number of edges labeled 0 and 1, respectively. A graph that admits such a labeling is termed a \emph{Fibonacci cordial graph}. In this paper, we investigate the existence and construction of Fibonacci cordial labelings for several families of graphs, including \emph{Generalized Petersen graphs}, \emph{open and closed helm graphs}, \emph{joint sum graphs}, and \emph{circulant graphs of small order}. New results and examples are presented, contributing to the growing body of knowledge on graph labelings inspired by numerical sequences.