# On $\gamma$-labelings of oriented graphs

Mathematica Bohemica (2007)

• Volume: 132, Issue: 2, page 185-203
• ISSN: 0862-7959

top Access to full text Full (PDF) Access to full text

## Abstract

top
Let $D$ be an oriented graph of order $n$ and size $m$. A $\gamma$-labeling of $D$ is a one-to-one function $f\phantom{\rule{0.222222em}{0ex}}V\left(D\right)\to \left\{0,1,2,...,m\right\}$ that induces a labeling ${f}^{\text{'}}\phantom{\rule{0.222222em}{0ex}}E\left(D\right)\to \left\{±1,±2,...,±m\right\}$ of the arcs of $D$ defined by ${f}^{\text{'}}\left(e\right)=f\left(v\right)-f\left(u\right)$ for each arc $e=\left(u,v\right)$ of $D$. The value of a $\gamma$-labeling $f$ is $\mathrm{v}al\left(f\right)={\sum }_{e\in E\left(G\right)}{f}^{\text{'}}\left(e\right).$ 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, $n\equiv 2\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}4\right)$, and every vertex of $G$ has odd degree.

## Citations in EuDML Documents

top

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.