Vertex-distinguishing VE-total Colorings of Cycles and Complete Graphs

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.