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.
@article{bwmeta1.element.doi-10_7151_dmgt_1659,
author = {Xiang'en Chen and Yuping Gao and Bing Yao},
title = {Vertex-Distinguishing IE-Total Colorings of Complete Bipartite Graphs Km,N(m < n)},
journal = {Discussiones Mathematicae Graph Theory},
volume = {33},
year = {2013},
pages = {289-306},
zbl = {1305.05071},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1659}
}
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) pp. 289-306. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1659/
[1] P.N. Balister, B. Bollobás and R.H. Shelp, Vertex distinguishing colorings of graphs with Δ(G) = 2, Discrete Math. 252 (2002) 17-29. doi:10.1016/S0012-365X(01)00287-4[Crossref]
[2] P.N. Balister, O.M. Riordan and R.H. Schelp, Vertex distinguishing edge colorings of graphs, J. Graph Theory 42 (2003) 95-109. doi:10.1002/jgt.10076[Crossref] | Zbl 1008.05067
[3] C. Bazgan, A. Harkat-Benhamdine, H. Li and M. Wo´zniak, On the vertexdistinguishing proper edge-colorings of graphs, J. Combin. Theory (B) 75 (1999) 288-301. doi:10.1006/jctb.1998.1884[Crossref]
[4] A.C. Burris and R.H. Schelp, Vertex-distinguishing proper edge-colorings, J. Graph Theory 26 (1997) 73-82. doi:10.1002/(SICI)1097-0118(199710)26:2h73::AID-JGT2i3.0.CO;2-C[Crossref] | Zbl 0886.05068
[5] J. Černý, M. Horňák and R. Soták, Observability of a graph, Math. Slovaca 46 (1996) 21-31. | Zbl 0853.05040
[6] X. Chen, Asymptotic behaviour of the vertex-distinguishing total chromatic numbers of n-cube, J. Northwest Univ. 41(5) (2005) 1-3. | Zbl 1092.05503
[7] F. Harary and M. Plantholt, The point-distinguishing chromatic index, in: F. Harary, J.S. Maybee (Eds.), Graphs and Application, New York (1985) 147-162.
[8] M. Horňák and R. Soták, Observability of complete multipartite graphs with equipotent parts, Ars Combin. 41 (1995) 289-301.
[9] M. Horňák and R. Soták, Asymptotic behaviour of the observability of Qn, Discrete Math. 176 (1997) 139-148. doi:10.1016/S0012-365X(96)00292-0[Crossref] | Zbl 0890.05028
[10] M. Horňák and R. Soták, The fifth jump of the point-distinguishing chromatic index of Kn,n, Ars Combin. 42 (1996) 233-242. | Zbl 0852.05045
[11] M. Horňák and R. Soták, Localization jumps of the point-distinguishing chromatic index of Kn,n, Discuss. Math. Graph Theory 17 (1997) 243-251. doi:10.7151/dmgt.1051[Crossref] | Zbl 0906.05025
[12] M. Horňák and N. Zagaglia Salvi, On the point-distinguishing chromatic index of complete bipartite graphs, Ars Combin. 80 (2006) 75-85.
[13] N. Zagaglia Salvi, On the point-distinguishing chromatic index of Kn,n, Ars Combin. 25B (1988) 93-104.
[14] N. Zagaglia Salvi, On the value of the point-distinguishing chromatic index of Kn,n, Ars Combin. 29B (1990) 235-244.
[15] Z. Zhang, P. Qiu, B. Xu, J. Li, X.Chen and B.Yao, Vertex-distinguishing total colorings of graphs, Ars Combin. 87 (2008) 33-45. | Zbl 1224.05203