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

Abstract:

      Let $G$ be a simple graph. Let $f$ be a mapping from $V(G)\cup E(G)$ to $\{1, 2,\cdots, k\}$. Let $C_f (v)=\{f(v)\}\cup \{f(vw)|w\in V(G), vw\in E(G)\}$ 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.