Edge-Odd Graceful Labeling on Circulant Graphs With Different Generating Sets

Abstract

Let $G = (V,E)$ be a simple, finite, undirected and connected graph. A graph $G =(V,E)$ with p vertices and q edges is said to have an edge - odd graceful labeling if there exists a bijection $f$ from $E$ to $\left\{ {1},{3},{5},\ldots.{2q}-{1} \right\}$ \linebreak so that the induced mapping ${{{f}}^{+}}~$ from V to $\left\{ 0,{1},{2},\ldots {2q}-{1} \right\}$ given by ${{{f}}^{+}}\left( {x} \right)={\sum}^{}\{f(xy/xy\in E\}\;\;mod\;|2q|)$. In this paper, we have constructed an edge тАУ odd graceful labeling on circulant graphs ${{{C}}_{{n}}}\left( {1},{2},{3},{4},{5},{6} \right)$ { and } ${{{C}}_{{n}}}\left( {1},{2},{3},{4},{5},{6},{7} \right)$ for odd $n$, ${n}\in {I}$. Here $(1,2,3,4,5,6)$ and $(1,2,3,4,5,6,7)$ are the generating sets.