Geodesic
On characterization of uniquely 3-list colorable complete multipartite graphs
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 1, pp. 105-114

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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: K_2,2,r r ∈ 4,5,6,7,8, K_2,3,4, K_1*4,4, K_1*4,5, K_1*5,4. Also, they conjectured that the nine graphs are not U3LC graphs. After that, except for K_2,2,r r ∈ 4,5,6,7,8, the others have been proved not to be U3LC graphs. In this paper we first prove that K_2,2,8 is not U3LC graph, and thus as a direct corollary, K_2,2,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 
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Zhao, Yancai; Shan, Erfang. On characterization of uniquely 3-list colorable complete multipartite graphs. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 1, pp. 105-114. http://geodesic.mathdoc.fr/item/DMGT_2010_30_1_a8/