Journal of Prime Research in Mathematics

# On the gracefulness of the digraphs $$n − C_{m}$$ for $$m$$ odd

Zhao Lingqi
College of Mathematics and Computer Science, Inner Mongolian University for Nationalities, 028043,P.R.China.
Jirimutu
College of Mathematics and Computer Science, Inner Mongolian University for Nationalities, 028043,P.R.China.
Xirong Xu
Department of Computer Science, Dalian University of Technology, 116024, P. R. China.
Wang Lei
Department of Computer Science, Dalian University of Technology, 116024, P. R. China.

$$^{1}$$Corresponding Author: jrmt@sina.com

### Abstract

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.

#### Keywords:

Digraph, directed cycles, graceful graph, graceful labeling.