Some Conclusion on Unique -List Colorable Complete Multipartite Graphs
If a graph admits a -list assignment such that has a unique -coloring, then is called uniquely -list colorable graph, or ULC graph for short. In the process of characterizing ULC graphs, the complete multipartite graphs are often researched. But it is usually not easy to construct the unique -list assignment of . In this paper, we give some propositions about the property of the graph when it is ULC, which provide a very significant guide for constructing such list assignment. Then a special example of ULC graphs as a application of these propositions is introduced. The conclusion will pave the way to characterize ULC complete multipartite graphs.
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 . Following the paper , 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  and Erdős et al. . For a graph and each vertex , let denote a list of colors available for . is said to be a list assignment of . If for all , then is called -list assignment of . For example, the numbers nearby the vertices in Figure 1 are -list assignment of the graph. A list coloring from a given collection of lists is a proper coloring such that is chosen from . We will refer to such a coloring as an L-coloring . 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 [6–8]. 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 . It is a discrete optimization problem. A model for channel availability observed by the secondary users is introduced in . 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 between uniquely list colorability and choosability of a graph are presented in . In this paper, we research the unique list colorability of graph.
The concept of unique list coloring was introduced by Dinitz and Martin  and independently by Mahmoodian and Mahdian , which can be used to study defining set of -coloring  and critical sets in Latin squares . Let be a graph with vertices, and suppose that for each vertex in , there exists a list of colors , 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 graph . That is to say that, a graph which has the property also has the property .
Mahdian and Mahmoodian  characterized uniquely -list colorable graphs. They showed the following.
Proposition 1 (see ). A connected graph has the property if and only if every block of is either a cycle, a complete graph, or a complete bipartite graph.
In paper , it is showed that recognizing uniquely -list colorable graphs is -complete for every ; then uniquely 3-list colorable graphs are unlikely to have a nice characterization. But Ghebleh and Mahmoodian  and He et al. [16–18] have characterized the ULC complete multipartite graphs, and one has the following.
Proposition 2 (see ). The graphs , , , , , , , , , and are ULC.
Wang et al.  have characterized ULC complete multipartite graphs with at least parts except for finitely many of them.
In the process of characterizing ULC complete multipartite graphs, it is often researched that the property of complete multipartite graphs has only one part whose size is more than one; that is, . Paper  studied the property of graphs and . The following was concluded.
Proposition 4 (see ). For every , .
The property of graphs and is researched in paper , and it is showed the following.
Proposition 5 (see ). For every , and have the property , and if , then .
Conclusions above are generalized by Wang et al.  recently.
Proposition 6 (see ). For every , , has the property .
Proposition 7 (see ). For every , , has the property .
But there is no other conclusion about what are the maximal numbers and such that the graph is a ULC graph for every . Besides, the property of list assignment of ULC graph is still unclear, and there is a lack of the necessary conditions for the ULC graph . It seems that the larger is, the more difficult the characterizing ULC graphs are.
In fact, if we want to proof that some graph 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 for every , not only for special .
In this paper the property of the graph 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 when it is a ULC graph. According to these propositions, a special example of ULC graphs 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.
2. Property of the ULC Graph
In this section, we list some theorems about the property of the graph 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 , if is a ULC graph, then ().
Proof. If , has the property by Proposition 1; so it is not a U2LC graph, nor a ULC graph which is contradictory to the suppose.
In view of these facts, it is supposed that for a ULC graph in the following.
In the process of proving Theorem 9, for convenience, the parts of are denoted by for and . For the given -list assignment : , , there is a unique -list color : , . Furthermore, one has , ; , ; ; .
Theorem 9. Suppose that , , and . Suppose that for some , , , is a ULC graph, and the unique -list color from the -list assignment is defined as above; then one has the following:(1), where and ; , where , ; (2); (3);(4), ;(5) and , ; (6)there must be a () such that .
Proof. Let .(1) From the definition of the ULC graph, this conclusion is obvious.(2) Suppose that , which means that . Let . We introduce a -list assignment to as follows. For every vertex in , if , then ; otherwise where and . Since induces a list coloring for , has exactly a -coloring, namely, the restriction of on . has the property 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.
Case 1. One has which means that are pairwise different.
Adding new edges between any two vertices in , the resulting graph is . Note that is also a proper -coloring of , and has the property 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.
Case 2. One has which means that in there are just two vertices assigned a common color, and the others are pairwise different.
Not loss of generality, we say that and are pairwise different for . Adding new edges between any two vertices in , the resulting graph is . A -list assignment to is introduced as follows. For every vertex in , if , then ; otherwise where and . It is obvious that for every , and the restriction of on is an -coloring of . Obviously, has the property by the Proposition 1 and the property . By the property of , we can obtain a new -coloring of , which can be extended to as follows. For every vertex in , if , then ; otherwise . From the construction of , it is obvious that is a new -coloring of which is contradictory to the fact that is the unique -list color.
In sum, .(4) If , then it is obvious that the conclusion is true. If , then we suppose that the conclusion is wrong, which means that there are two numbers and such that and . It is clear that . Let and let for but . Obviously, is a new -coloring of which is contradictory to the fact that is the unique -list color.(5) Proof by contradiction. Suppose that , and there are and such that , and . Let and let for but . Obviously is a new -coloring of which is contradictory to the fact that is the unique -list color. Then from (3) we know that , .(6) By contradiction. Suppose for every () that ; then it is obtained that . So we get that for every . Let . We introduce a -list assignment to as follows. For every vertex in , we obtain by randomly getting rid of elements from such that and , as can be done because . Since induces a list coloring for , has exactly one -coloring, namely, the restriction of on . has the property by Proposition 1; so 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. An Example of ULC Graphs
According the property of ULC graph in Theorem 9, we construct a list assignment of graph for special and and prove that the graph is a ULC graph in this section.
Theorem 10. Let where , the graph is a ULC graph.
Proof. For convenience, the () parts of are denoted by for and .
Let . We denote all -subsets of by , where . Now consider with the following -list of colors on vertices. For , . For every , , .
For example, when , , and the above mentioned -list assignment for is as follows:
Note that the list assignment makes a total of colors. Since is a complete -partite graph, the last part can take colors at the most. From the construction of , it is obtained obviously that is the unique choice for a -list coloring from . Then the must take the color . In the example above, the colors in the list coloring are marked by underlines. So a unique -list coloring from is made and is a ULC graph.
Notice that for every , , has the property by Proposition 7. Now the graph in Theorem 10 is a ULC graph; so it is wrong with the proposition “for every , , has the property ”. Therefore, () is the maximal numbers in Proposition 7.
4. Discussion and Some Open Problems
It is not easy to characterize ULC complete multipartite for any . In fact, it is a very tricky job to construct a -list assignment such that there exists a unique list coloring. Theorem 9 provides a direction for constructing such list assignment of , and perhaps it makes construction easier for the researchers. Furthermore, Theorem 9 is true for every and , and the conclusion is extensive.
It must be noted that the conditions in Theorem 9 are only necessary conditions of for ULC graph, not sufficient conditions.
Theorem 10 can be regarded as a application of Theorem 9. And from Theorem 10, it is known that () is exactly the maximal numbers in Proposition 7. But notice that the number is not the minimal number for every in Proposition 7. For example, when , and is ULC according to Theorem 10. In fact is ULC by Propositions 2 and 3; so is not the minimal number for in Proposition 7. Moreover, it is very likely that for different the minimal number is different in Proposition 7.
The following problem arises naturally from the work.
Problem. For every , characterize all minimum number such that the graph is a ULC graph.
This research was supported by the National Natural Science Foundation of China (No. 61271409), the China Postdoctoral Science Foundation (No. 2012M510768, No. 2013T60264), and a grant from the Science and Technology Research and Development Program of Qinhuangdao (No. 201001A151).
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