Abstract
In this paper, the -coloring problem of the graph is studied with application to channel allocation of the wireless network. First, by introducing two new logical operators, some necessary and sufficient conditions for solving the -coloring problem are given. Moreover, it is noted that all solutions of the obtained logical equations are corresponding to each coloring scheme. Second, by using the semitensor product, the necessary and sufficient conditions are converted to an algebraic form. Based on this, all coloring schemes can be obtained through searching all column indices of the zero columns. Finally, the obtained result is applied to analyze channel allocation of the wireless network. Furthermore, an illustration example is given to show the effectiveness of the obtained results in this paper.
1. Introduction
It is well known that the coloring problem is a basic and classical problem in graph theory. Graph coloring is originated from famous conjecture called four-colour conjecture [1] and widely used in many real-life areas [2–4], such as scheduling and timetabling in engineering, air traffic flow management, and channel allocation of mobiles. There are various forms of graph coloring, such as set coloring, list coloring, -coloring, and -coloring ( denotes a nonnegative integer, ). The labeling problems of graphs arise in many networking and telecommunication contexts. The channel allocation problem is first formulated as a graph coloring problem by Hale [5]. Furthermore, Griggs and Yeh formulated this problem as a graph labeling problem [6]. -label coloring is one kind of graph labeling, which has major application in channel allocation [5, 7–10]. Thus, it is still more interesting to introduce a new method to study the coloring problem.
Recently, Cheng et al. and Li et al. provided a new mathematical method, which is called the semitensor product with matrices [11–13] to study logical systems [14–23], probability logical networks [24, 25], game theory [26, 27], coloring problem [1, 10, 28], and some other related fields [29–31]. Wang et al. first studied the graph problem by using the semitensor product [1]. In [1], the maximum (weight) stable set and vertex coloring problems of graphs were investigated with application to the group consensus of multiagent systems, and an algorithm was established to find all the internally stable sets for any graph. In [28], Zhong et al. investigated the minimum stable set and core of the graph and established an algorithm to find all the externally stable sets.
This paper studies the -coloring problem of the graph with application to channel allocation of the wireless network. Some necessary and sufficient conditions for solving the -coloring problem are first made by introducing two new logical operators. Moreover, it is noted that all solutions of the obtained logical equations are corresponding to each coloring scheme. Then, by using the semitensor product, the necessary and sufficient conditions are converted to an algebraic form. Based on this, all coloring schemes can be obtained through searching all column indices of the zero columns. Finally, the obtained results are applied to analyze channel allocation of the wireless network. Furthermore, an illustration example is given to show the effectiveness of the obtained results in this paper.
The rest of this paper is organized as follows. Section 2 gives some necessary preliminaries on the semitensor product of matrices and -labeling. The main results are shown in Section 3. In Section 4, we apply the obtained results to the channel allocation of the wireless network, which is followed by the conclusion in Section 5.
2. Preliminaries
In this section, we give some necessary preliminaries on the semitensor product, the pseudo-Boolean function, and graph theory, which will be used in the sequel.
First, we give some notations to be used in this paper. , especially, . : the -th column of the identity matrix . Denote by the -th column of matrix and by the set of all columns of matrix . : the set of real matrices, where denotes the set of real numbers. , and for simplicity, let . Identify , , which implies , where means they are equivalent. A matrix is called a logical matrix if columns of are of the form of . Denote by the set of logical matrices. If , it can be expressed as . For the sake of compactness, it is briefly denoted by .
Next, we give some definitions and results about the semitensor product.
Definition 1. (see [11]). The semitensor product of two matrices and iswhere is the least common multiple of and and is the Kronecker product.
Throughout this paper, the default matrix product is the semitensor product. The semitensor product is a generalization of the conventional matrix product. Thus, we can simply call it “product” and omit the symbol “” without confusion.
Definition 2. (see [11]). A swap matrix is an matrix defined as follows: its rows and columns are labeled by double index , the columns are arranged by the ordered multiindex , and the rows are arranged by the ordered multiindex . Then, the element at the position is
Remark 1. When , is briefly rewritten as . Furthermore, from Definition 2, can be written as the following form for all , :Now, we list some basic properties of the semitensor product [11]:(1)Let and be column vectors. Then,(2)Let be a column vector. Then,(3)Let be a logical vector. Then, where(4)Let be a logical vector and . Then,where is the -th block of . Especially, when ,
Lemma 1 .(see [11]). Any logical function with logical variables , , can be expressed in a multilinear form aswhere and is unique, called the structural matrix of .
Remark 2. The first row of the structural matrix corresponds to the truth value of the logical function .
Now, we list the structural matrices for some basic -valued logical operators [11], which will be used later.
Negation (): , which has a structural matrix as .
Conjunction (): , which has a structural matrix asDisjunction (): , which has a structural matrix asExclusive or (): , which has a structural matrix asDummy operator (): , which has a structural matrix asThe following concepts and properties will be used in the next section.
Definition 3. (see [12]). An -ary pseudo-Boolean function is a mapping from to , where .
A graph consists of a vertex (node) set and an edge set denoted by .
Lemma 2 .(see [32]). Given a simple graph , an -labeling of is an integer assignment such thatwhere , denotes the distance between and , and are two given positive integers.
3. Main Results
In this section, we investigate the -label coloring problem by the semitensor product method and present the main results of this paper.
Consider a graph with nodes . Assume that the adjacency matrix, , of is given aswhere denotes a neighbor set of node .
It is noted that for an undirected graph and in our study since the graph is a simple graph. Furthermore, let , where and , and are, respectively, Boolean addition and Boolean multiplication. It is easy to obtain that when and when .
Let . For all , assign it an integer , i.e., . We need two logical operators as
Moreover, the structural matrices are
Then, we have the following result to determine whether the -label problem is solved.
Theorem 1. Consider an undirected graph with nodes . Its -label problem is solved if and only if the logical equationsare solved. That is,is solved.
Proof. Necessity: assume that coloring of is solvable, and is the adjacent matrix of the graph . For all and , when , . It is easy to see that and , . That is, when or when .
Since , we introduce the logical operators and , and then we have is equivalent to , and is equivalent to . That is, is equivalent toThen, we have for all . Since for the undirected graph and , we obtain thatSet , where and , . Obviously, . Similarly, when , we haveTherefore, from (22) and (23), we obtain that (19) is satisfied, and the necessity is proved.
Sufficiency: suppose that (19) is satisfied. Then, for all , we haveFrom (24), if , . Then, or , i.e., , . Therefore, . Similarly, if , by (24).
Since and , we have the coloring of the graph is solvable, and the proof is complete.
It is note that, for a directed graph , we have the following corollary.
Corollary 1. Consider a directed graph with nodes . Its -label problem is solved if and only ifis solved.
Let , . Then, . Using the semitensor product and the vector form of logical variables, we have the following results.
Theorem 2. Logical equations (19) are solved if and only if there exists at least an integer such that the -th column of matrix is 0, where, and the product is
Proof. Using the semitensor product and the vector form of logical variables, there exists one matrix such that the left-hand side of equation (13) is , where , . Equation (13) is solved if and only if there exists at least an integer such that the column of matrix is 0. Now, we only need to study matrix .
Since the logical form of iswe havewhereFurthermore,where , ,and the product is
Therefore,Thus, logical equations (19) are solved if and only if there exists such thatthat is, the -th column of is zero. Then, the proof is complete.
Based on Theorem 2, we give the following algorithm to find all for the coloring solutions of the given graph.Using formula (24), compute and . SetThen, all -label coloring plans of are .
4. Illustrative Example
In this section, we give an example to illustrate the effectiveness of the results/algorithms obtained in this paper.
In order to avoid interference with each other, different channels need to be assigned to different base stations in the wireless network. Moreover, the main object of the channel allocation problem is to search an allocation scheme which has the channel as least as possible. Some mathematical models can be used to study the channel allocation problem of the wireless network, including -coloring, list coloring, set coloring, and -coloring. The most commonly used model is -coloring. Denote by the topological graph of the wireless network, where is a node set denoting base stations or their users and denotes an edge set. Now, we will use the -coloring model to analyze the channel allocation of the wireless network.
Example 1. Construct the telecommunication base stations among four cities denoted by , respectively. Denote a city by one vertex of the graph. If the base station constructed in city can cover city , then there is an edge between and . Now, the covering graph of base stations is established as shown in Figure 1. Our target is to search all schemes for channel allocation of the wireless network. In the wireless network, the channel interval is greater than or equal to 2 for two adjacent base stations and greater than or equal to 1 for two base stations with distance 2.
From Figure 1, we have its adjacency matrix as the following:According to , we have by the above definitionUsing Algorithm 1, we can compute the matrix defined in (26) by the semitensor product. Then, by Matlab, different channels which are denoted by 0, 1, 2, 3, and 4 are needed to satisfy the requirement in the channel allocation of the wireless network. Moreover, all detailed schemes are corresponding toThe corresponding channel schemes are shown in Table 1.
For example, for , . That is, channels 4, 1, 0, and 3 are, respectively, assigned to stations A, B, C, and D which satisfies the requirement in the channel allocation of the wireless network. Moreover, from the table, there are at least 4 channels to satisfy the graph.
5. Conclusion
The -coloring problem of the graph is studied with application to channel allocation of the wireless network in this paper. The necessary and sufficient conditions for solving the -coloring problem are given by introducing two new logical operators. Moreover, it is noted that all solutions of the obtained logical equations are corresponding to each coloring scheme. By using the semitensor product, the necessary and sufficient conditions are converted to an algebraic form. Based on this, all coloring schemes can be obtained through searching all column indices of the zero columns. Furthermore, an illustration example on channel allocation of the wireless network is given to show the effectiveness of the obtained results in this paper. In future, we plan to study other coloring problems by using the semitensor product method, i.e., -coloring, list coloring, and set coloring.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (G61673243, G69674001, and G61403223), the Postdoctoral Science Foundation Funded Project of China (2017M612234), and the Research Fund for High-Level Personnel of Qingdao Agricultural University (631426).