Vertex-distinguishing IE-total colorings of complete bipartite graphs K_{m,n}(m

Let G be a simple graph.An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color.Let C(u) be the set of colors of vertex u and edges incident to u under f .For an IE-total coloring f of G using k colors, if C(u) = C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G, or a k-VDIET coloring of G for short.The minimum number of colors required for a VDIET coloring of G is denoted by χ ie vt (G), and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short.VDIET colorings of complete bipartite graphs K m,n (m < n) are discussed in this paper.Particularly, the VDIET chromatic numbers of K m,n (1 ≤ m ≤ 7, m < n) as well as complete graphs K n are obtained. Keywords: complete bipartite graphs, IE-total coloring, vertex-distinguishing IE-total coloring, vertex-distinguishing IE-total chromatic number.2010 Mathematics Subject Classification: 05C15.From [15] we know that the above conjecture is valid for complete graphs, complete bipartite graphs, paths and cycles, etc.In this paper we propose a kind of vertex-distinguishing general total coloring called IE-total coloring.The relationship between this coloring and vertexdistinguishing proper total coloring is similar to the relationship between vertexdistinguishing general edge coloring and vertex-distinguishing proper edge coloring.An IE-total coloring of a graph G is a total coloring of G such that the Condition (v) is satisfied.If f is an IE-total coloring of graph G using k colors