@article {IOPORT.06084663,
    author       = {Feng, Wei},
    title        = {On the gracefulness of the digraphs $n-\vec{C}_{m}$.},
    year         = {2012},
    journal      = {International Journal of Pure and Applied Mathematics},
    volume       = {78},
    number       = {3},
    issn         = {1311-8080},
    pages        = {429-434},
    publisher    = {Academic Publications, Sofia},
    abstract     = {Summary: 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 $f^{'}(u,v)=[ f(v)-f(u) ]$ (mod $(\vert E\vert+1)$ 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 $n$.},
    identifier   = {06084663},
}