On the gracefulness of the digraphs \(n − C_{m}\) for \(m\) odd

A digraph D(V, E) is said to be graceful if there exists an injection \(f : V (G) → {0, 1, · · · , |E|}\) such that the induced function \(f’:E(G) → {1, 2, · · · , |E|}\) which is defined by \(f'(u, v) = [f(v)−f(u)] (mod |E|+1)\) for every directed edge \((u, v)\) is a bijection. Here, \(f\) is called a graceful labeling (graceful numbering) of \(D(V, E)\), while \(f’\) is called the induced edge’s graceful labeling of D. In this paper we discuss the gracefulness of the digraph \(n − C_{m}\) and prove that \(n − C_{m}\) is a graceful digraph for \(m = 5, 7, 9, 11, 13\) and even n.