Vertex-Distinguishing E-Total Coloring of the Graphs mC3 and mC4

Let G be a simple graph.A total coloring f of G is called E-total-coloring if no two adjacent vertices of G receive the same color and no edge of G receives the same color as one of its endpoints.For E-total-coloring f of a graph G and any vertex u of G,let Cf(u) or C(u) denote the set of colors of vertex u and the edges incident to u.We call C(u) the color set of u.If C(u) = C(v) for any two different vertices u and v of V(G),then we say that f is a vertex-distinguishing E-total-coloring of G,or a V DET coloring of G for short.The minimum number of colors required for a V DET colorings of G is denoted by χevt(G),and it is called the VDET chromatic number of G.In this article,we will discuss vertex-distinguishing E-total colorings of the graphs mC3 and mC4.