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: ${\displaystyle K_{2,2,r}}$ r ∈ {4,5,6,7,8}, ${\displaystyle K_{2,3,4}}$, ${\displaystyle K_{1\cdot 4,4}}$, ${\displaystyle K_{1\cdot 4,5}}$, ${\displaystyle K_{1\cdot 5,4}}$. Also, they conjectured that the nine graphs are not U3LC graphs. After that, except for ${\displaystyle K_{2,2,r}}$ r ∈ {4,5,6,7,8}, the others have been proved not to be U3LC graphs. In this paper we first prove that ${\displaystyle K_{2,2,8}}$ is not U3LC graph, and thus as a direct corollary, ${\displaystyle K_{2,2,r}}$ (r = 4,5,6,7,8) are not U3LC graphs, and then the uniquely 3-list colorable complete multipartite graphs are characterized completely.