On γ -labelings of oriented graphs

Futaba Okamoto; Ping Zhang; Varaporn Saenpholphat

Mathematica Bohemica (2007)

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

Abstract

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Let D be an oriented graph of order n and size m . A γ -labeling of D is a one-to-one function f V ( D ) { 0 , 1 , 2 , ... , m } that induces a labeling f ' E ( D ) { ± 1 , ± 2 , ... , ± m } of the arcs of D defined by f ' ( e ) = f ( v ) - f ( u ) for each arc e = ( u , v ) of D . The value of a γ -labeling f is v a l ( f ) = e E ( G ) f ' ( e ) . A γ -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 2 is orientably balanced unless G is a tree, n 2 ( m o d 4 ) , and every vertex of G has odd degree.

How to cite

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Okamoto, Futaba, Zhang, Ping, and Saenpholphat, Varaporn. "On $\gamma $-labelings of oriented graphs." Mathematica Bohemica 132.2 (2007): 185-203. <http://eudml.org/doc/250253>.

@article{Okamoto2007,
abstract = {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 $\mathop \{\mathrm \{v\}al\}(f) = \sum _\{e \in E(G)\} f^\{\prime \}(e).$ 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.},
author = {Okamoto, Futaba, Zhang, Ping, Saenpholphat, Varaporn},
journal = {Mathematica Bohemica},
keywords = {oriented graph; $\gamma $-labeling; balanced $\gamma $-labeling; balanced oriented graph; orientably balanced graph; balanced -labeling; balanced oriented graph; orientably balanced graph},
language = {eng},
number = {2},
pages = {185-203},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On $\gamma $-labelings of oriented graphs},
url = {http://eudml.org/doc/250253},
volume = {132},
year = {2007},
}

TY - JOUR
AU - Okamoto, Futaba
AU - Zhang, Ping
AU - Saenpholphat, Varaporn
TI - On $\gamma $-labelings of oriented graphs
JO - Mathematica Bohemica
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 132
IS - 2
SP - 185
EP - 203
AB - 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 $\mathop {\mathrm {v}al}(f) = \sum _{e \in E(G)} f^{\prime }(e).$ 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.
LA - eng
KW - oriented graph; $\gamma $-labeling; balanced $\gamma $-labeling; balanced oriented graph; orientably balanced graph; balanced -labeling; balanced oriented graph; orientably balanced graph
UR - http://eudml.org/doc/250253
ER -

References

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  2. On γ -labelings of trees, Discuss. Math., Graph Theory 25 (2005), 363–383. (2005) MR2233002
  3. Introduction to Graph Theory, McGraw-Hill, Boston, 2005. (2005) 
  4. A dynamic survey of graph labeling, Electron. J. Combin. 5 (1998), Dynamic Survey 6, pp. 43. (1998) Zbl0953.05067MR1668059
  5. How to number a graph, Graph Theory Comp. Academic Press, New York, 1972, pp. 23–37. (1972) Zbl0293.05150MR0340107
  6. On certain valuations of the vertices of a graph, Theory Graphs, Proc. Int. Symp. Rome 1966, Gordon and Breach, New York, 1967, pp. 349–355. (1967) Zbl0193.53204MR0223271

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