VertexDistinguishing ETotal Coloring of the Graphs $mC_{3}$ and $mC_{4}$ 
Received:January 01, 2009 Revised:January 28, 2010 
Key Word:
coloring Etotal coloring vertexdistinguishing Etotal coloring vertexdistinguishing Etotal chromatic number the vertexdisjoint union of $m$ cycles with length $n$.

Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.10771091) and the Scientific Research Project of Northwest Normal University (Grant No.NWNUKJCXGC0361). 

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Abstract: 
Let $G$ be a simple graph. A total coloring $f$ of $G$ is called Etotalcoloring if no two adjacent vertices of $G$ receive the same color and no edge of $G$ receives the same color as one of its endpoints. For Etotalcoloring $f$ of a graph $G$ and any vertex $u$ of $G$, let $C_f(u)$ or $C(u)$ denote the set of colors of vertex $u$ and the edges incident to $u$. We call $C(u)$ the color set of $u$. If $C(u)\neq C(v)$ for any two different vertices $u$ and $v$ of $V(G)$, then we say that $f$ is a vertexdistinguishing Etotalcoloring of $G$, or a $VDET$ coloring of $G$ for short. The minimum number of colors required for a $VDET$ colorings of $G$ is denoted by $\chi_{vt}^e(G)$, and it is called the VDET chromatic number of $G$. In this article, we will discuss vertexdistinguishing Etotal colorings of the graphs $mC_{3}$ and $mC_{4}$. 
Citation: 
DOI:10.3770/j.issn:1000341X.2011.01.005 
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