Odd graceful labeling for the jewel graph and the extended jewel graph without the prime edge

Abstract

R.B. Gnanajothi (Topics in Graph Theory, Ph.D. Thesis, Madurai Kamaraj University, Madurai, Tamil Nadu, India, 1991) introduced odd graceful labeling. A function $ f $ is called an odd graceful labeling of a graph $ G $ if $ f : V (G) \rightarrow \{{0, 1, 2, ...,2q-1\}} $ is injective and the induced function $ f^*: E(G) \rightarrow \{{1, 3, . . . , 2q - 1\}} $ defined as $ f^*(e = uv) =|f(u)-f(v)| $ is bijective. A graph which admits an odd graceful labeling is called an odd graceful graph. Many results exist on odd graceful labeling. The concept of odd graceful labeling is implemented in the areas of coding theory.

In this paper we prove that the jewel graph $ J_{n}^* $ and the extended jewel graph $EJ_{n,m}^* $ without the prime edge is odd graceful.