Vertex-Distinguishing IE-Total Colorings of Complete Bipartite Graphs Km,N(m n)
Discussiones Mathematicae Graph Theory, Tome 33 (2013) no. 2, p. 289.

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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) 6= 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 χie 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 Km,n(m < n) are discussed in this paper. Particularly, the VDIET chromatic numbers of Km,n(1 ≤ m ≤ 7,m < n) as well as complete graphs Kn are obtained.
Classification : 05C15
Mots-clés : complete bipartite graphs, IE-total coloring, vertex-distinguishing IE-total coloring, vertex-distinguishing IE-total chromatic number 
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Xiang’en Chen; Yuping Gao; Bing Yao. Vertex-Distinguishing IE-Total Colorings of Complete Bipartite Graphs Km,N(m < n). Discussiones Mathematicae Graph Theory, Tome 33 (2013) no. 2, p. 289. https://geodesic-test.mathdoc.fr/item/DMGT_2013__33_2_267543/