Farey Graceful Labeling of Graphs and Its Applications

Motivated by the work of Kumar et al. (On Farey graceful labeling, 2023) that introduced labeling of graphs particularly, paths, cycles, hairy cycles and caterpillars using Farey sequence \(F_n\) $F_n$ of order n, for \(n \in \textbf{N}\) $n\in \mathbf{N}$ , defined by the set of numbers \(\frac{l}{m} \in Q\) $\frac{l}{m}\in Q$ with \(0 \le l \le m \le n\) $0\le l\le m\le n$ and l, m are co-prime placed in increasing order and this labeling was called Farey graceful labeling, we extend this concept to wider range of graphs and show that the graphs \(P_2\times P_n\) $P_2\times P_n$ , complete bipartite graphs, firecracker trees are all Farey graceful.