$P_m\times K_n$的邻点可区别全色数

设 $G$ 是简单图. 设$f$是一个从$V(G)\cup E(G)$ 到$\{1, 2,\cdots, k\}$的映射. 对每个$v\in V(G)$, 令 $C_f (v)=\{f(v)\}\cup \{f(vw)|w\in V(G), vw\in E(G)\}$. 如果 $f$是$k$-正常全染色, 且对任意$u, v\in V(G), uv\in E(G)$, 有$C_f(u)\ne C_f(v)$, 那么称 $f$ 为图$G$的邻点可区别全染色(简称为$k$-AVDTC).数 $\chi_{at}(G)=\min\{k|G$ 有$k$-AVDTC\}称为图$G$的邻点可区别全色数.本文给出路$P_m$和完全图$K_n$ 的Cartesion积的邻点可区别全色数.

Adjacent-Vertex-Distinguishing Total Chromatic Number of $P_m\times K_n$

Let $G$ be a simple graph. Let $f$ be a mapping from $V(G)\cup E(G)$ to $\{1, 2,\cdots, k\}$. Let $C_f (v)=\{f(v)\}\cup \{f(vw)|w\in V(G), vw\in E(G)\}$ for every $v\in V(G)$. If $f$ is a $k$-proper-total-coloring, and if $C_f(u)\ne C_f(v)$ for $u, v\in V(G), uv\in E(G)$, then $f$ is called $k$-adjacent-vertex-distinguishing total coloring of $G$($k$-AVDTC of $G$ for short). Let $\chi_{at}(G)=\min\{k|G$ 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 the Cartesion product of path $P_m$ and complete graph $K_n$ is obtained.