Discussiones Mathematicae Graph Theory 30(1) (2010) 105-114
doi: 10.7151/dmgt.1480

[BIBTex] [PDF] [PS]

ON CHARACTERIZATION OF UNIQUELY 3-LIST COLORABLE COMPLETE MULTIPARTITE GRAPHS

Yancai Zhao1,2  and  Erfang Shan1

1Department of Mathematics
Shanghai University
Shanghai 200444, P.R. China
2Department of Science
Bengbu University
Anhui 233030, P.R. China

e-mail: zhaoyc69@126.com

Abstract

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.

Keywords: list coloring, complete multipartite graph, uniquely 3-list colorable graph.

2010 Mathematics Subject Classification: 05C15.

References

[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.
[2] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (American Elsevier Publishing Co., INC., New York, 1976).
[3] J.H. Dinitz and W.J. Martin, The stipulation polynomial of a uniquely list colorable graph, Austral. J. Combin. 11 (1995) 105-115.
[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.
[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.
[6] M. Ghebleh and E.S. Mahmoodian, On uniquely list colorable graphs, Ars Combin. 59 (2001) 307-318.
[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.
[8] M. Mahdian and E.S. Mahmoodian, A characterization of uniquely 2-list colorable graphs, Ars Combin. 51 (1999) 295-305.
[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.
[10] Y.F. Shen and Y.N. Wang, On uniquely list colorable complete multipartite graphs, Ars Combin. 88 (2008) 367-377.
[11] V.G. Vizing, Coloring the vertices of a graph in prescribed colors, (in Russian) Discret. Anal. 29 (1976) 3-10.
[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: 10.1007/978-3-540-70666-3_30.

Received 26 February 2009
Revised 2 April 2009
Accepted 14 April 2009