Adjacent vertex distinguishing total coloring on P_m∨P_n

Let G be a simple connected graph.A k-proper total coloring of G is called adjacent-distinguishing if for arbitrary two adjacent vertices u and v,C(u)≠C(v),where C(u) is the set of the colors of u and edges which is adjacent to u.The minimum k such that G has a k-adjacent-vertex-distinguishing total coloring is called the adjacent vertex distinguishing total chromatic number.The adjacent vertex distinguishing total chromatic number is obtained for the graph formed by two paths.