Radio Graceful Labelling of Graphs

Abstract

Radio labelling problem of graphs have their roots in communication problem known as \emph{Channel Assignment Problem}. For a simple connected graph $G=(V(G), E(G))$, a radio labeling is a mapping $f \colon V(G)\rightarrow \{0,1,2,\ldots\}$ such that $|f(u)-f(v)|\geq {\rm diam}(G)+1-d(u,v)$ for each pair of distinct vertices $u,v\in V(G)$, where $\rm{diam}(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$. A radio labeling $f$ of a graph $G$ is a \emph{radio graceful labeling} of $G$ if $f(V(G)) = \{0,1,\ldots, |V(G)|-1\}$. A graph for which a radio graceful labeling exists is called \emph{radio graceful}. In this article, we study radio graceful labeling for general graphs in terms of some new parameters.

Keywords

• channel assignment problem
• radio labelling
• radio graceful labelling
• radio number
• span