Adjacent-Vertex-Distinguishing Total Chromatic Number of Pm × Kn

Let G be a simple graph. Let f be a mapping from V (G)∪E(G) to {1, 2, · · · , k}. Let Cf (v) = {f(v)} ∪ {f(vw)|w ∈ V (G), vw ∈ E(G)} for every v ∈ V (G). If f is a k-propertotal-coloring, and if Cf (u) 6= Cf (v) for u, v ∈ V (G), uv ∈ E(G), then f is called k-adjacentvertex-distinguishing total coloring of G(k-AVDTC of G for short). Let χat(G) = min{k|G has a k-adjacent-vertex-distinguishing total coloring}. Then χat(G) is called the adjacent-vertexdistinguishing total chromatic number. The adjacent-vertex-distinguishing total chromatic number on the Cartesion product of path Pm and complete graph Kn is obtained.