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Keywords:
oriented graph; $\gamma$-labeling; balanced $\gamma$-labeling; balanced oriented graph; orientably balanced graph
Summary:
Let $D$ be an oriented graph of order $n$ and size $m$. A $\gamma$-labeling of $D$ is a one-to-one function $f\: V(D) \rightarrow \lbrace 0, 1, 2, \ldots , m\rbrace$ that induces a labeling $f^{\prime }\: E(D) \rightarrow \lbrace \pm 1, \pm 2, \ldots , \pm m\rbrace$ of the arcs of $D$ defined by $f^{\prime }(e) = f(v)-f(u)$ for each arc $e =(u, v)$ of $D$. The value of a $\gamma$-labeling $f$ is $\mathop {\mathrm val}(f) = \sum _{e \in E(G)} f^{\prime }(e).$ A $\gamma$-labeling of $D$ is balanced if the value of $f$ is 0. An oriented graph $D$ is balanced if $D$ has a balanced labeling. A graph $G$ is orientably balanced if $G$ has a balanced orientation. It is shown that a connected graph $G$ of order $n \ge 2$ is orientably balanced unless $G$ is a tree, $n \equiv 2 \hspace{4.44443pt}(\@mod \; 4)$, and every vertex of $G$ has odd degree.
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