| 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 x of G,let C(x)denote the set of colors of vertex x and of the edges incident with x,we call C(x)the color set of x.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 VDET coloring of G for short.The minimum number of colors required for a VDET coloring of G is denoted by Xe vt(G)and is called the VDET chromatic number of G.The VDET coloring of complete bipartite graph Km,n(m = 3,4,5)is discussed in this paper and the VDET chromatic number of Km,n(m= 3,4,5)has been obtained. |
