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

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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


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.


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