Odd graceful labeling in cycle with extended bistar

Abstract

Graph  labeling  creates a new direction towards  research areas in  graph  theory and  has  various applications  in coding  approach, communication  networks and  many  more.  In 1991,  Gnanajothi (Topics in Graph Theory, Ph.D. Thesis, Madurai Kamraj University, Tamil Nadu, India) introduced a labeling  method  called odd graceful  labeling.   A graph $G$ with $q$ edges is odd graceful  if there is an injection,  $f :V(G) \to \{0,1,2,\ldots,(2q-1)\}$  such that when each $xy$ edge is assigned the label $\left|f\left(x\right)-f(y)\right|$, the resulting edge labels are $\{1,3,5,\ldots, (2q-1)\}$. In this paper, we prove the odd graceful labeling in a cycle with extended bistar.