Some conclusion on unique k-list colorable complete multipartite graphs

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 uklc graph for short. in the process of characterizing uklc graphs,the complete multipartite graphs k1r,s(r,sn) are often researched. but it is usually not easy to construct the unique k-list assignment of k1r,s. in this paper,we give some propositions about the property of the graph k1r,s when it is uklc,which provide a very significant guide for constructing such list assignment. then a special example of uklc graphs k1r,s as a application of these propositions is introduced. the conclusion will pave the way to characterize uklc complete multipartite graphs. © 2013 yanning wang et al.