$mC_{3}\vee nC_{3}$ 和 $mC_{4}\vee nC_{4}$的点可区别I-全染色及VI-全染色
Vertex-Distinguishing I-Total coloring and VI-Total coloring of $mC_{3}\vee nC_{3}$ and $mC_{4}\vee nC_{4}$

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 作者 单位 E-mail 陈祥恩 西北师范大学 chenxe@nwnu.edu.cn 张生桂 西北师范大学

设 $f$ 为图 $G$ 的一个一般全染色(即 若干种颜色对图$G$的全部顶点及边的一个分配), 如果任意两个相邻点染以不同颜色且任意两条相邻边染以不同的颜色，则称为图 $G$ 的I-全染色; 如果任意两条相邻边染以不同的颜色，则称为图 $G$ 的VI-全染色. 用 $C(x)$ 表示在 $f$ 下点 $x$ 的颜色以及与 $x$ 关联的边的色所构成的集合(非多重集). 对图 $G$ 的一个I-全染色(分别地，VI- 全染色) $f$, 一旦 $\forall u, v\in V(G), u\neq v$, 就有 $C(u)\neq C(v)$, 则 $f$ 称为图 $G$ 的点可区别的I-全染色(或点可区别VI-全染色), 简称为VDIT染色(分别地，VDVIT染色). 令 $\chi _{vt}^{i}(G)=\min\{k| G \mbox{存在} k-\mbox{VDIT染色}\}.$ 称 $\chi _{vt}^{i}(G)$ 为图 $G$ 的点可区别I-全色数. 令 $\chi _{vt}^{vi}(G)=\min\{k| G \mbox{存在} k-\mbox{VDVIT染色}\}.$ 称 $\chi _{vt}^{vi}(G)$ 为图 $G$ 的点可区别VI-全色数. 在本文利用构造具体的染色的方法，讨论了联图$mC_{3}\vee nC_{3}$和 $mC_{4}\vee nC_{4}$ 的点可区别I-全染色和点可区别VI- 全染色并给出了联图$mC_{3}\vee nC_{3}$和 $mC_{4}\vee nC_{4}$ 的点可区别I-全色数和点可区别VI- 全色数.

Let $G$ be a simple graph. Supose $f$ is a general total coloring of $G$ (i.e., an assignment of several colors to all vertices and edges of $G$ ), if any two adjacent vertices and any two adjacent edges of $G$ are assigned different colors, then $f$ is called an I-total coloring of a graph $G$; if any two adjacent edges of $G$ are assigned different colors, then $f$ is called a VI-total coloring of a graph $G$. Let $C(x)$ denote the set of colors of vertex $x$ and of the edges incident with $x$ under $f$. For an I-total coloring(resp. , VI-total coloring ) $f$ of a graph $G$, if $C(u)\neq C(v)$ for any two distinct vertices $u$ and $v$ of $V(G)$, then $f$ is called a vertex distinguishing I-total coloring (resp., vertex distinguishing VI-total coloring )of $G$. Let $\chi _{vt}^{i}(G)=\min\{k|\text{G has a$k$-VDIT coloring}\}.$ Then $\chi _{vt}^{i}(G)$ is called the VDIT chromatic number of $G$. Let $\chi _{vt}^{vi}(G)=\min\{k| \text{G has a$k$-VDVIT coloring}\}.$ Then $\chi _{vt}^{vi}(G)$ is called the VDVIT chromatic number of $G$. In this parper, we discussed the VDIT coloring (resp., VDVIT coloring) of $mC_{3}\vee nC_{3}$ and $mC_{4}\vee nC_{4}$ and determined the VDIT chromatic number (resp., VDVIT chromatic number) of them by constructing concrete coloring.
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