IJPAM: Volume 78, No. 3 (2012)

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

Wei Feng
College of Mathematics
Inner Mongolian University for Nationalities
Tongliao, 028043, P.R. CHINA


Abstract. A digraph $ D(V,E)$ is said to be graceful if there exists an injection $f:V(D) \rightarrow \{0, 1, \cdots, \vert E\vert \}$ such that the induced function $f':E(D) \rightarrow \{1, 2, \cdots, \vert E\vert \}$ 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 digraph $ D(V,E)$, and $f'$ 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 the digraph $n-\vec{C}_{21}$ is graceful for even $n$.

Received: April 1, 2012

AMS Subject Classification: 05C65

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

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 78
Issue: 3