Adjacent-Vertex-Distinguishing Total Chromatic Number of P_m×K_n

Let G be a simple graph.Let f be a mapping from V(G)∪E(G) to {1,2,…,k}. Let C_f(v)={f(v)}∪{f(vw)|w∈V(G),vw∈E(G)}for every v∈V(G).If f is a k-proper- total-coloring,and if C_f(u)≠C_f(v) for u,v∈V(G),uv∈E(G),then f is called k-adjacent- vertex-distinguishing total coloring of G(k-AVDTC of G for short).Let x_(at)(G)=min{k|G has a k-adjacent-vertex-distinguishing total coloring}.Then x_(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.