On the gracefulness of the digraphs n − C m for m odd

Abstract

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