Odd gracefulness of cycle with subdivided shell graphs

Abstract

 In 1991 Gnanajothi (Gnanajothi, R.B. (1991). Topics in Graph Theory, Ph.D. Thesis, Madurai Kamaraj University, Tamil Nadu, India) introduced a labeling method called odd graceful labeling. A graph  $G$  with $q$ edges  is  said  to  be  odd  graceful  if  there  is  an injection $\ f\ :V\ (G)\ \to \ \{0,\ 1,\ 2,\ \dots ,\ \left(2q-1\right)\}$, such that when each edge $\ xy\ \ $ is assigned the label $\left|\ f\left(x\right)-\ f\left(y\right)\right|,$ the resulting edges labels are distinct and they are members of the set $\left\{1,3,5,\ldots ,\left(2q-1\right)\right\}$. In this paper, we prove that the graph obtained by attaching each vertex of $C_m\ $ with the subdivided shell graph is odd graceful, when $\ m\equiv 0\left({\rm mod\ 4}\right)$.