Graceful Vit Labeling: A New Approach and Its Applications to Graphs

Abstract

Consider an undirected, simple graph \( G = (V(G), E(G)) \). A graceful labeling of graph \( G \) is an injective function \(f: V(G) \to \{0, 1, 2, \dots, |E(G)|\} \) such that the induced edge labels, defined by $f^*(uv)=|f(u)-f(v)|$ for every edge $uv \in E(G)$, are all distinct. In this paper, we introduce a new type of graceful labeling, called \textbf{graceful vit labeling}. A graceful labeling $f$ of graph $G$ is called a graceful vit labeling if the vertex weight function $w_f: V(G) \to \mathbb{N}$, defined by $w_f(v)=f(v)+\sum_{uv \in E(G)}f^*(uv)$ for every $v \in V(G)$, assigns pairwise distinct weights to all vertices in graph $G$. In other words, no two vertices have the same sum of their own label and the labels of all edges incident to them. In this paper we present various examples of graphs that admit graceful vit labeling and explores structural properties and necessary conditions for their existence.