Some conclusion on unique \(k\)-list colorable complete multipartite graphs (Q1789876)

Summary: If a graph \(G\) admits a \(k\)-list assignment \(L\) such that \(G\) has a unique \(L\)-coloring, then \(G\) is called uniquely \(k\)-list colorable graph, or U\(k\)LC graph for short. In the process of characterizing U\(k\)LC graphs, the complete multipartite graphs \(K_{1*r,s}\) (\(r,s\in N\)) are often researched. But it is usually not easy to construct the unique \(k\)-list assignment of \(K_{1*r,s}\). In this paper, we give some propositions about the property of the graph \(K_{1*r,s}\) when it is U\(k\)LC, which provide a very significant guide for constructing such list assignment. Then a special example of U\(k\)LC graphs \(K_{1*r,s}\) as a application of these propositions is introduced. The conclusion will pave the way to characterize U\(k\)LC complete multipartite graphs.