Some Conclusion on Unique k-List Colorable Complete Multipartite Graphs

Abstract

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 $K_{1\mathrm{*}r,s}(r,s\in N)$ are often researched. But it is usually not easy to construct the unique $k$-list assignment of $K_{1\mathrm{*}r,s}$. In this paper, we give some propositions about the property of the graph $K_{1\mathrm{*}r,s}$ 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 $K_{1\mathrm{*}r,s}$ as a application of these propositions is introduced. The conclusion will pave the way to characterize U$k$LC complete multipartite graphs.