On the Gracefulness of the Digraphs \(n – \overrightarrow{C}_m\)

Abstract

A digraph \(D(V, E)\) is said to be graceful if there exists an injection \(f: V(G) \to \{0, 1, \ldots, |E|\}\) such that the induced function \(f’: E(G) \to \{1, 2, \ldots, |E|\}\) which is defined by \(f'(u, v) = [f(v) – f(u)] \pmod{|E| + 1}\) 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 \(n – \overrightarrow{C}_m\), and prove that \(n – \overrightarrow{C}_m\) is a graceful digraph for \(m = 4, 6, 8, 10\) and even \(n\).