IJPAM: Volume 23, No. 3 (2005)

ON THE GRACEFULNESS OF THE DIGRAPHS $n\cdot\vec{C}_{m}$

Jirimutu$^1$Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024, P.R. CHINA, Jun Wang$^2$, Xu Xirong$^3$
$^{1,2}$Department of Applied Mathematics
Dalian University of Technology
Dalian, 116024, P.R. CHINA
$^1$e-mail: jrmt@sina.com
$^2$e-mail: junwang@.dlut.edu.cn
$^{1}$College of Mathematics and Computer Science
Inner Mongolian University for Nationalities
Tongliao 028043, P.R. CHINA
$^3$Department of Computer Science
Dalian Nationalities University
Dalian, 116600, P.R. CHINA
e-mail: xuxirong2002@163.com


Abstract.A digraph $D(V,E)$ is said to be graceful if there exists an injection $ f:V(G) \rightarrow \{0,1, \cdots,\vert E\vert\}$ such that the induced function $f^{'}: E(G) \rightarrow \{1, 2, \cdots, \vert E\vert\}$ which is defined by $ f^{'}(u,v)=[ f(v)-f(u) ]\pmod {\vert E\vert+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 $n\cdot\vec{C}_{m}$ and prove that $n\cdot\vec{C}_{m}$ is a graceful digraph for $m=9,11,13$ and even $n$.

Received: July 22, 2005

AMS Subject Classification: 05C65

Key Words and Phrases: digraph, directed cycles, graceful graph, graceful labeling

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 23
Issue: 3