Vertex-distinguishing total coloring of graphs

Let G be a simple and connected graph of order p >= 2. A proper k-total-coloring of a graph G is a mapping f from V(G) U E(G) into {1, 2, center dot center dot center dot, k} such that every two adjacent or incident elements of V(G) boolean OR E(G) are assigned different colors. Let C-f (u)=f(u) boolean OR{f(uv) vertical bar uv is an element of E(G)} be the neighbor color-set of u, if C-f (u)not equal C-f (v) for any two vertices u and v of V(G), we say f a vertex-distinguishing proper k-total-coloring of G, or a k-VDT-coloring of G for short. The minimal number of all over k-VDT-colorings of G is denoted by chi(vt)(G), and it is called the VDTC chromatic number of G. For some special families of the complete graph K, complete bipartite graph K-m,K-n, path P-m and circle C-m, etc., we get their VDTC chromatic numbers and propose a conjecture in this article.