On The Gracefulness of The Digraphs \(n-\vec{C}_m\)

Abstract

A digraph \(D(V, E)\) is said to be graceful if there exists an injection \(f : V(D) \rightarrow \{0, 1, \ldots, |V|\}\) such that the induced function \(f’ : E(D) \rightarrow \{1, 2, \ldots, |V|\}\) 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 digraph \(D(V, E)\), while \(f’\) is called the induced edge’s graceful labeling of digraph \(D(V,E)\). In this paper, we discuss the gracefulness of the digraph \(n-\vec{C}_m\) and prove that the digraph \(n-\vec{C}_{17}\) is graceful for even \(n\).