Geodesic
Vertex-Distinguishing IE-Total Colorings of Complete Bipartite Graphs Km,N(m < n)
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 289-306

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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) ≠ 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 χievt(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 lt; n) are discussed in this paper. Particularly, the VDIET chromatic numbers of Km,n(1 ≤ m ≤ 7, m lt; n) as well as complete graphs Kn are obtained.
Keywords: complete bipartite graphs, IE-total coloring, vertex-distinguishing IE-total coloring, vertex-distinguishing IE-total chromatic number 
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     title = {Vertex-Distinguishing {IE-Total} {Colorings} of {Complete} {Bipartite} {Graphs} {K\protect\textsubscript{m,N}(m} &lt; n)},
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Chen, Xiang’en; Gao, Yuping; Yao, Bing. Vertex-Distinguishing IE-Total Colorings of Complete Bipartite Graphs Km,N(m < n). Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 289-306. http://geodesic.mathdoc.fr/item/DMGT_2013_33_2_a3/