Graceful distance labeling for some particular graphs

Abstract

In this paper, we define a new type of labeling for graphs which we call graceful distance labeling (GDL). An injective mapping $f$ from the vertex set $V(G)$ into the set of non-negative integers such that the absolute difference of labels of vertices $u$ and $v$ is greater than or equal to distance between them i.e. $|f(u)-f(v)| \geq d(u, v)$ where $d(u, v)$ denotes the distance between the vertices $u$ and $v$ in $G$. The graceful distance labeling number (GDLN), $\lambda_d(G)$ of $G$ is the minimum $k$ where $G$ has a graceful distance labeling $f$ with $k$ being the absolute difference between the largest and smallest image points of $f$ i.e. $\lambda_d(G)=\min k$, where $k=\max |f(u)-f(v)|$. In this paper, we find the values of $k$ for different graphs.