On characterization of uniquely 3-list colorable complete multipartite graphs
Yancai Zhao ; Erfang Shan
Discussiones Mathematicae Graph Theory, Tome 30 (2010), p. 105-114 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

For each vertex v of a graph G, if there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Ghebleh and Mahmoodian characterized uniquely 3-list colorable complete multipartite graphs except for nine graphs: K2,2,r r ∈ 4,5,6,7,8, K2,3,4, K1*4,4, K1*4,5, K1*5,4. Also, they conjectured that the nine graphs are not U3LC graphs. After that, except for K2,2,r r ∈ 4,5,6,7,8, the others have been proved not to be U3LC graphs. In this paper we first prove that K2,2,8 is not U3LC graph, and thus as a direct corollary, K2,2,r (r = 4,5,6,7,8) are not U3LC graphs, and then the uniquely 3-list colorable complete multipartite graphs are characterized completely.

Publié le : 2010-01-01
EUDML-ID : urn:eudml:doc:270997 
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1480,
     author = {Yancai Zhao and Erfang Shan},
     title = {On characterization of uniquely 3-list colorable complete multipartite graphs},
     journal = {Discussiones Mathematicae Graph Theory},
     volume = {30},
     year = {2010},
     pages = {105-114},
     zbl = {1215.05078},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1480}
}

Yancai Zhao; Erfang Shan. On characterization of uniquely 3-list colorable complete multipartite graphs. Discussiones Mathematicae Graph Theory, Tome 30 (2010) pp. 105-114. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1480/

[000] [1] N. Alon, Restricted colorings of graphs, in: K. Walker, editor, Surveys in Combinatorics, Number 187 in London Math. Soc. LNS, pp. 1-33, 1993. | Zbl 0791.05034

[001] [2] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (American Elsevier Publishing Co., INC., New York, 1976). | Zbl 1226.05083

[002] [3] J.H. Dinitz and W.J. Martin, The stipulation polynomial of a uniquely list colorable graph, Austral. J. Combin. 11 (1995) 105-115. | Zbl 0826.05025

[003] [4] P. Erdös, A.L. Rubin and H. Taylor, Choosability in graphs, in: Proceedings of West Coast Conference on Combinatorics, Graph Theory and Computing, number 26 in Congr. Number., pp. 125-157, Arcata, CA, September 1979.

[004] [5] Y.G. Ganjali, M. Ghebleh, H. Hajiabohassan, M. Mirzadeh and B.S. Sadjad, Uniquely 2-list colorable graphs, Discrete Appl. Math. 119 (2002) 217-225, doi: 10.1016/S0166-218X(00)00335-8. | Zbl 0999.05037

[005] [6] M. Ghebleh and E.S. Mahmoodian, On uniquely list colorable graphs, Ars Combin. 59 (2001) 307-318. | Zbl 1066.05063

[006] [7] W.J. He, Y.N. Wang, Y.F. Shen and X. Ma, On property M(3) of some complete multipartite graphs, Australasian Journal of Combinatorics, to appear. | Zbl 1097.05019

[007] [8] M. Mahdian and E.S. Mahmoodian, A characterization of uniquely 2-list colorable graphs, Ars Combin. 51 (1999) 295-305. | Zbl 0977.05046

[008] [9] E.S. Mahmoodian and M. Mahdian, On the uniquely list colorable graphs, in: Proceedings of the 28th Annual Iranian Mathematics Conference, Part 1, number 377 in Tabriz Univ. Ser., Tabriz, 1997. | Zbl 0977.05046

[009] [10] Y.F. Shen and Y.N. Wang, On uniquely list colorable complete multipartite graphs, Ars Combin. 88 (2008) 367-377. | Zbl 1224.05188

[010] [11] V.G. Vizing, Coloring the vertices of a graph in prescribed colors, (in Russian) Discret. Anal. 29 (1976) 3-10.

[011] [12] Y.Q. Zhao, W.J. He, Y.F. Shen and Y.N. Wang, Note on characterization of uniquely 3-list colorable complete multipartite graphs, in: Discrete Geometry, Combinatorics and Graph Theory, LNCS 4381 (Springer, Berlin, 2007) 278-287, doi: 10.1007/978-3-540-70666-3₃0. | Zbl 1149.05312