Edge \(\delta\)-graceful labeling for some cyclic-related graphs (Q2308040)

Summary: In this paper, we introduce a new type of labeling of a graph \(G\) with \(p\) vertices and \(q\) edges called edge \(\delta\)-graceful labeling, for any positive integer \(\delta\), as a bijective mapping \(f\) of the edge set \(E (G)\) into the set \(\{\delta, 2 \delta, 3 \delta, \dots, q \delta\}\) such that the induced mapping \(f^\ast : V(G) \to \{0, \delta, 2 \delta, 3 \delta, \dots, q \delta - \delta\}\), given by \(f^\ast(u) = (\sum_{u v \in E (G)} f (uv)) \mod (\delta k)\), where \(k = \max (p, q)\), is an injective function. We prove the existence of an edge \(\delta\)-graceful labeling, for any positive integer \(\delta\), for some cycle-related graphs like the wheel graph, alternate triangular cycle, double wheel graph \(W_{n, n}\), the prism graph \(\Pi_n\), the prism of the wheel \(P(W_n)\), the gear graph \(G_n\), the closed helm CH\(_n\), the butterfly graph \(B_n\), and the friendship \(\mathrm{Fr}_n\).