IJPAM: Volume 36, No. 2 (2007)

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

Yulan Bao$^1$, Xirong Xu$^2$, Jirimutu$^3$
$^{1,3}$College of Mathematics and Computer Science
Inner Mongolia University for Nationalities
Tongliao, 028043, P.R. CHINA
$^3$e-mail: jrmt@sina.com
$^2$Department of Mathematics
University of Science and Technology of China
Hefei, 230026, P.R. CHINA
e-mail: xirongxu@ustc.edu.cn


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 $m=19$ and even $n$.

Received: February 2, 2007

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: 2007
Volume: 36
Issue: 2