Hilbert Graceful Labeling on the Eight Sprocket Graph

Abstract

Let \( G \) be a simple, finite, connected, undirected, non-trivial graph with \( p \) vertices and \( q \) edges. Let \( V(G) \) be the vertex set and \( E(G) \) be the edge set of \( G \). The \( n^{\text{th}} \) Hilbert number is denoted by \( H_n \) and is defined by $H_n = 4(n - 1) + 1, \quad \text{where } n \ge 1$. A \emph{Hilbert graceful labeling} is an injective function $H : V(G) \rightarrow \{x : x = 4(i - 1) + 1, \, 1 \le i \le 2q\}$ which induces a bijective function $H^* : E(G) \rightarrow \{1, 2, 3, 4, \ldots, q\}$ defined by $H^*(uv) = \frac{1}{4} \, |H(u) - H(v)|$,  $\forall \, uv \in E(G), \, u, v \in V(G)$. A graph that admits a Hilbert graceful labeling is called a \emph{Hilbert graceful graph}. This paper focuses on the Eight Sprocket Graph \( SC_n \) and demonstrates its Hilbert gracefulness. It also investigates related graph families formed from copies of \( SC_n \), proving that the path union, cycle, and star of the Eight Sprocket Graph are all Hilbert graceful.