| A Note on Adjacent-Vertex-Distinguishing Total Chromatic Numbers for $P_m\times P_n, P_m\times C_n$ and $C_m\times C_n$ |
| Received:September 12, 2006 Revised:October 28, 2007 |
| Key Words:
total coloring adjacent-vertex-distinguishing total coloring adjacent-vertex-distinguishing total chromatic number.
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| Fund Project:the National Natural Science Foundation of China (No.10771091); the Science and Research Project of the Education Department of Gansu Province (No.0501-02). |
| Author Name | Affiliation | | CHEN Xiang En | College of Mathematics and Information Science, Northwest Normal University, Gansu 730070, China | | ZHANG Zhong Fu | College of Mathematics and Information Science, Northwest Normal University, Gansu 730070, China
Institute of Applied Mathematics, Lanzhou Jiaotong University, Gansu 730070, China | | SUN Yi Rong | College of Mathematics and Information Science, Northwest Normal University, Gansu 730070, China |
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| Abstract: |
| Let $G$ be a simple graph. Let $f$ be a mapping from $V(G) \cup E(G)$ to $\{1, 2,\ldots, k\}$. Let $C_f(v)=\{f(v)\}\cup \{f (vw)|w\in V(G), vw\in E(G)\}$ for every $v\in V(G)$. If $f$ is a $k$-proper-total-coloring, and for $u, v\in V(G), uv\in E(G)$, we have $C_f(u)\neq C_f(v)$, then $f$ is called a $k$-adjacent-vertex-distinguishing total coloring ($k$-$AVDTC$ for short). Let $\chi_{at} (G)= \min\{k|G$ have a $k$-adjacent-vertex-distinguishing total coloring$\}$. Then $\chi_{at} (G)$ is called the adjacent-vertex-distinguishing total chromatic number ($AVDTC$ number for short). The $AVDTC$ numbers for $P_m\times P_n, P_m\times C_n$ and $C_m\times C_n$ are obtained in this paper. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2008.04.006 |
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