The ϕ-labeling on Certain Classes of Graphs

This paper presents a comprehensive study of Fibonacci range labelings, also known as ϕ-labelings, for graphs constructed via binary operations, with a principal focus on Cartesian products and graph unions. A ϕ-labeling is an injective assignment of Fibonacci numbers to the vertices of a graph such that the induced edge labels, defined by a specific rational function, are all distinct and the sequence of their successive ratios converges to the golden ratio, ϕ. We establish that the Cartesian product of paths and complete graphs, Pn×Km for n ≥ 3,m ≥ 2, admits such a labeling. Furthermore, we prove that the union of any two cycles, Cn∪Cm for n,m ≥ 3, is also a ϕ-labeling graph. These results generalize and extend the foundational work on ladder graphs and star products. The proofs are constructive, providing explicit labeling functions and a rigorous analytical verification of the convergence of the edge label ratios to ϕ. The study concludes by discussing the constraints of this labeling scheme, particularly its limitation to graphs with at most two central vertices labeled with the smallest Fibonacci numbers.