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 K 1∗r,s (r, s ∈ 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 graphK 1∗r,s when it is UkLC, which provide a very significant guide for constructing such list assignment. Then a special example of UkLC graphsK 1∗r,s as a application of these propositions is introduced.The conclusion will pave the way to characterize UkLC complete multipartite graphs.


Introduction
In this section, some definitions and results about list colorings which are referred to throughout the paper are introduced.For the necessary definitions and notation, we refer the reader to standard texts, such as [1].Following the paper [2], we use the notation   *  (,  ∈ ,  is the set of natural numbers) for a complete -partite graph in which each part is of size .Notation such as   * , , (, ,  ∈ ) is used similarly.
The idea of list colorings of graphs is due, independently, to Vizing [3] and Erdős et al. [4].For a graph  = (, ) and each vertex V ∈ (), let (V) denote a list of colors available for V.  = {(V) | V ∈ ()} is said to be a list assignment of .If |(V)| =  for all V ∈ (), then  is called -list assignment of .For example, the numbers nearby the vertices in Figure 1 are 2-list assignment of the graph.A list coloring from a given collection of lists is a proper coloring  such that (V) is chosen from (V).We will refer to such a coloring as an L-coloring [5].In Figure 1, the set of circled numbers makes a 2-list coloring of the graph.
The list coloring model can be used in the channel assignment [6][7][8].The fixed channel allocation scheme leads to low channel utilization across the whole channel.It requires a more effective channel assignment and management policy, which allows unused parts of channel to become available temporarily for other usages so that the scarcity of the channel can be largely mitigated [6].It is a discrete optimization problem.A model for channel availability observed by the secondary users is introduced in [6].We abstract each secondary network topology into a graph, where vertices represent wireless users such as wireless lines, WLANs, or cells, and edges represent interferences between vertices.In particular, if two vertices are connected by an edge in the graph, we assume that these two vertices cannot use the same spectrum simultaneously.In addition, we associate with each vertex a set, which represents the available spectra at this location.Due to the differences in the geographical location of each vertex, the sets of spectra of different nodes may be different.Then a list coloring model is constructed.
The research of list coloring consists of two parts: the choosability and the unique list colorability.Some relations (2, 3) between uniquely list colorability and choosability of a graph are presented in [9].In this paper, we research the unique list colorability of graph.The concept of unique list coloring was introduced by Dinitz and Martin [10] and independently by Mahmoodian and Mahdian [11], which can be used to study defining set of -coloring [12] and critical sets in Latin squares [13].Let  be a graph with  vertices, and suppose that for each vertex V in , there exists a list of  colors (V), such that there exists a unique -coloring for ; then  is called uniquely k-list colorable graph or a ULC graph for short.It is obvious that the set of circled numbers makes a 2-list coloring of the graph in Figure 2.For a graph , it is said to have the property () if and only if it is not uniquely -list colorable graph.So  has the property () if for any collection of lists assigned to its vertices, each of size , either there is no list coloring for  or there exist two list colorings.Note that the -number of a graph , denoted by (), is defined to be the least integer  such that  has the property ().
It is clear from the definition of uniquely -list colorable graphs that each ULC graph is also a U( − 1)LC graph [14].That is to say that, a graph which has the property ( − 1) also has the property ().
Proposition 3 (see [16]).Let  be a complete multipartite graph; then  is U3LC if and only if it has one of the graphs in Proposition 2 as an induced subgraph.Wang et al. [19] have characterized U4LC complete multipartite graphs with at least 6 parts except for finitely many of them.
Conclusions above are generalized by Wang et al. [20] recently.
But there is no other conclusion about what are the maximal numbers  and  such that the graph  1 * , is a ULC graph for every .Besides, the property of list assignment of ULC graph  1 * , is still unclear, and there is a lack of the necessary conditions for the ULC graph  1 * , .It seems that the larger  is, the more difficult the characterizing ULC graphs are.
In fact, if we want to proof that some graph  1 * , is a ULC graph, we must find a -list assignment such that there exists a unique list coloring.In general it is not easy to construct such list assignment, and it usually requires a lot of skills.But if some properties of such graphs are known, the construction process perhaps will become easier.In addition, it is hoped that one can obtain some properties of ULC graph  1 * , for every , not only for special .
In this paper the property of the graph  1 * , is researched when it is a ULC graph.The paper is organized as follows.In Section 2, we give some propositions about the property of the graph  1 * , when it is a ULC graph.According to these propositions, a special example of ULC graphs  1 * , is introduced in Section 3. In Section 4, we discuss the results and give an open problem.The conclusion will pave the way to characterize ULC complete multipartite graphs.

Property of the U𝑘LC Graph 𝐾 1 * 𝑟,𝑠
In this section, we list some theorems about the property of the graph  1 * , when it is ULC, as it is conducive to construct the list assignment of ULC complete multipartite graphs and characterize the ULC graphs.Theorem 8.For every  ≥ 1, if  1 * , is a ULC graph, then  ≥ 2 (,  ∈ ).
In view of these facts, it is supposed that  ≥ 2 for a ULC graph  1 * , in the following.
(1) From the definition of the ULC graph, this conclusion is obvious.
(2) Suppose that Let  =  −  +1 =  1 *  .We introduce a ( − 1)-list assignment   to  as follows.For every vertex V in , if where  ∈ (V) and  ̸ = (V).Since  induces a list coloring  for ,  has exactly a   -coloring, namely, the restriction of  on . =  1 *  has the property (2) by Proposition 1; so it has the property (), and we can obtain a new   -coloring of .From the construction of   , we know that the new   -coloring can be extended to .Thus,  has a new -coloring which is different from  which is contradictory to the fact that  is the unique -list color.
(3) We use the reduction to absurdity.Adding new edges between any two vertices in {V +1 , V +2 , . . ., V + }, the resulting graph is   =  + .Note that  is also a proper -coloring of   , and   has the property (2) by Proposition 1; hence   has the property ().So we can obtain another coloring of   , which is also a legal -coloring for , which is contradictory to the fact that  is the unique -list color.

Figure 1 :
Figure 1: The list of graph.

Figure 2 :
Figure 2: The list coloring of graph.