Vertex-Distinguishing E-Total Colorings of Graphs

Let G be a simple graph. A total coloring f of G is called an 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 an E-total coloring f of a graph G and any vertex u of G let C f (u) or C(u) denote the set of colors of vertex u and of the edges incident to u. We call C(u) the color set of u. If C(u) not equal 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 VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by chi(e)(vl)(G) and is called the VDET chromatic number of G. In this paper, we find the VDET chromatic number for some special families of graphs, such as the path P-n, the cycle C-n, the complete bipartite graphs K-1,K-n and K-2, n the complete graph K-n, and wheels and fans, and we also propose a related conjecture.