Note on Characterization of Uniquely 3-List Colorable Complete Multipartite Graphs

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.