 Vertex-Distinguishing E-Total Coloring of the Graphs $mC_{3}$ and $mC_{4}$
Received:January 01, 2009  Revised:January 28, 2010
Key Word: coloring   E-total coloring   vertex-distinguishing E-total coloring   vertex-distinguishing E-total chromatic number   the vertex-disjoint 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.NWNU-KJCXGC-03-61).
 Author Name Affiliation Xiang En CHEN College of Mathematics and Information Science, Northwest Normal University, Gansu 730070, P. R. China Yue ZU College of Mathematics and Information Science, Northwest Normal University, Gansu 730070, P. R. China
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Let $G$ be a simple graph. A total coloring $f$ of $G$ is called E-total-coloring 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 E-total-coloring $f$ 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 vertex-distinguishing E-total-coloring 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 vertex-distinguishing E-total colorings of the graphs $mC_{3}$ and $mC_{4}$.