Abstract: Let G be a simple graph of order at least 2. A VE-total-coloring using k colors of a graph G is a mapping f from V (G) E(G) into {1,2,···,k} such that no edge receives the same color as one of its endpoints. Let C(u)={f(u)} {f(uv) | uv ∈ E(G)} be the color-set of u. If C(u)=C(v) for any two vertices u and v of V (G), then f is called a k-vertex-distinguishing VE-total coloring of G or a k-VDVET coloring of G for short. The minimum number of colors required for a VDVET coloring of G is denoted by χ^ve_vt(G) and it is called the VDVET chromatic number of G. In this paper we get cycle C_n, path P_n and complete graph K_n of their VDVET chromatic numbers and propose a related conjecture. 

Key words: graphs, VE-total coloring, vertex-distinguishing VE-total coloring, vertex-distinguishing  VE-total chromatic number