Radio Graceful Labelling of Graphs

Abstract

Radio labelling problem of graphs have its roots in communication problem known as Channel Assignment Problem . For a simple connected graph G = ( V ( G ) ; E ( G )), a radio labeling is a mapping f : V ( G ) → { 0 ; 1 ; 2 ; : : : } such that | f ( u ) − f ( v ) | ≥ diam( G )+ 1 − d ( u; v ) for each pair of distinct vertices u; v ∈ V ( G ), where 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 radio graceful labeling of G if f ( V ( G )) = { 0 ; 1 ; : : : ; | V ( G ) | − 1 } . A graph for which a radio graceful labeling exists is called radio graceful . In this article, a necessary and sufficient condition for radio graceful graphs are presented. Also some consequences of radio graceful graphs are given in terms of some new graph parameters.