IJPAM: Volume 18, No. 4 (2005)

ADJACENT-VERTEX-DISTINGUISHING TOTAL
CHROMATIC NUMBERS ON MONO-CYCLE
GRAPHS AND THE SQUARE OF CYCLES

Xiang-en Chen$^1$, Zhong-fu Zhang$^2$, Yi-rong Sun$^3$
$^{1,3}$College of Mathematics and Information Science
Northwest Normal University
Lanzhou, 730070, P.R. CHINA
$^1$e-mail: chenxe@nwnu.edu.cn
$^2$Institute of Applied Mathematics
Lanzhou Jiaotong University
Lanzhou, 730070, P.R. CHINA


Abstract.Let $G$ be a simple graph. Let $f$ be a mapping from $V(G)\cup E(G)$ to $\{1, 2,\cdots, k\}$. Let $C_f (v)=\{f(v)\}\cup \{f(vw)\vert w\in V(G), vw\in E(G)\}$ for every $v\in V(G)$. If $f$ is $k$-proper-total-coloring, and for $u, v\in V(G), uv\in E(G)$, we have that $C_f(u)\ne C_f(v)$, then $f$ is called the $k$-adjacent-vertex-distinguishing total coloring ($k-AVDTC$ for short). Let $\chi_{at}(G)=\min\{k\vert G$ has a $k$-adjacent-vertex-distinguishing total coloring}. Then $\chi_{at}(G)$ is called the adjacent-vertex-distinguishing total chromatic number. The adjacent-vertex-distinguishing total chromatic number on mono-cycle graphs and the square of cycles are obtained in this paper.

Received: December 17, 2004

AMS Subject Classification: 05C75

Key Words and Phrases: graph, total coloring, adjacent-vertex-distinguishing total coloring, adjacent-vertex-distinguishing total chromatic number

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 18
Issue: 4