L(p, q)-Label Coloring Problem with Application to Channel Allocation

In this paper, the L(p, q)-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 L(p, q)-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.


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][3][4], such as scheduling and timetabling in engineering, air traffic flow management, and channel allocation of mobiles. ere are various forms of graph coloring, such as set coloring, list coloring, T-coloring, and L(n 1 , n 2 , · · · , n s )-coloring (n i denotes a nonnegative integer, i � 1, 2, · · · , s). e labeling problems of graphs arise in many networking and telecommunication contexts. e 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]. L(p, q)-label coloring is one kind of graph labeling, which has major application in channel allocation [5,[7][8][9][10]. us, 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][12][13] to study logical systems [14][15][16][17][18][19][20][21][22][23], probability logical networks [24,25], game theory [26,27], coloring problem [1,10,28], and some other related fields [29][30][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.
is paper studies the L(p, q)-coloring problem of the graph with application to channel allocation of the wireless network. Some necessary and sufficient conditions for solving the L(p, q)-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. en, 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. e rest of this paper is organized as follows. Section 2 gives some necessary preliminaries on the semitensor product of matrices and L(p, q)-labeling. e 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.

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. R n×r : the set of n × r real matrices, where R denotes the set of real numbers. Δ n : � δ i n | i � 1, 2, · · · , n , and for simplicity, let where p ∼ q means they are equivalent. A matrix L ∈ R n×r is called a logical matrix if columns of L are of the form of δ i n . Denote by L n×r the set of n × r logical matrices.
If L ∈ L n×r , it can be expressed as L � [δ i 1 n δ i 2 n · · · δ i r n ]. For the sake of compactness, it is briefly denoted by L � δ n [i 1 i 2 · · · i r ].
Next, we give some definitions and results about the semitensor product.
Definition 1 (see [11]). e semitensor product of two matrices A ∈ R m×n and B ∈ R p×q is where α � lcm(n, p) is the least common multiple of n and p and ⊗ is the Kronecker product. roughout this paper, the default matrix product is the semitensor product. e semitensor product is a generalization of the conventional matrix product. us, we can simply call it "product" and omit the symbol " < imes" without confusion.
Definition 2 (see [11]). A swap matrix W [m,n] is an mn × mn matrix defined as follows: its rows and columns are labeled by double index (i, j), the columns are arranged by the ordered multiindex I d(i, j; m, n), and the rows are arranged by the ordered multiindex I d(j, i; n, m). en, the element at the position [(I, J), (i, j)] is Now, we list some basic properties of the semitensor product [11]: (1) Let X ∈ R m and Y ∈ R n be column vectors. en, (2) Let X ∈ R t be a column vector. en, (3) Let X � δ i t ∈ R t be a logical vector. en, where (4) Let X � δ i t ∈ R t be a logical vector and A ∈ R m×nt . en, Especially, when n � 1, Lemma 1 (see [11]). Any logical function n, can be expressed in a multilinear form as where y ∈ Δ k and M f ∈ L k×k n is unique, called the structural matrix of f.

Remark 2.
e first row of the structural matrix M f corresponds to the truth value of the logical function f(x 1 , x 2 , · · · , x s ). Now, we list the structural matrices for some basic k-valued logical operators [11], which will be used later.
Conjunction (∧): x∧y � min x, y , which has a structural matrix as Disjunction (∨): x∨y � max x, y , which has a structural matrix as Exclusive or (∨ ): x∨ y � 1, x � y 0, x ≠ y , which has a structural matrix as Dummy operator (σ d ): σ d x 2 � x, which has a structural matrix as e following concepts and properties will be used in the next section.

Lemma 2 (see [32]). Given a simple graph G � (V, E), an L(p, q)-labeling f of G is an integer assignment
where u, v ∈ V, d(u, v) denotes the distance between u and v, and p, q are two given positive integers.

Main Results
In this section, we investigate the L(p, q)-label coloring problem by the semitensor product method and present the main results of this paper. Consider a graph G with n nodes V � v 1 , v 2 , · · · , v n . Assume that the adjacency matrix, A � [a ij ], of G is given as where N i denotes a neighbor set of node i. It is noted that a ij � a ji for an undirected graph and a ii � 0 in our study since the graph G is a simple graph. Furthermore, are, respectively, Boolean addition and Boolean multiplication. It is easy to obtain that d(v i , v j ) � 1 when a ij � 1 and For all v i ∈ V, assign it an integer y i ∈ S, i.e., f(v i ) � y i . We need two logical operators as Moreover, the structural matrices are en, we have the following result to determine whether the L(p, q)-label problem is solved.

q)-label problem is solved if and only if the logical equations
Mathematical Problems in Engineering are solved. at is, is solved.
Proof. Necessity: assume that L(p, q) coloring of G is solvable, and A � [a ij ] is the adjacent matrix of the graph G.
at is, y i ≥ y j + p when y i ≥ y j or y i ≤ y j − p when y i ≤ y j . Since y i ∈ 0, 1, · · · , k − 1 { }, we introduce the logical operators ⊘ k+ and ⊘ k− , and then we have Since a ij � a ji for the undirected graph and |y i − y j | � |y j − y i |, we obtain that erefore, from (22) and (23), we obtain that (19) is satisfied, and the necessity is proved.
Since a ij � 1⇔d(v i , v j ) � 1 and b ij � 1⇔d(v i , v j ) � 2, we have the L(p, q) coloring of the graph G is solvable, and the proof is complete.
It is note that, for a directed graph G, we have the following corollary.
is solved. 1, 0]x. 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
, and the product is " ∝ " Proof. Using the semitensor product and the vector form of logical variables, there exists one matrix M ∈ R 1×k n such that the left-hand side of equation (13) is Mx 1 x 2 · · · x n , where (13) is solved if and only if there exists at least an integer i(1 ≤ i ≤ k n ) such that the ith column of matrix M is 0. Now, we only need to study matrix M.
Since the logical form of we have where Furthermore, Mathematical Problems in Engineering 5 where x � x 1 x 2 · · · x n , M ij,p � M p M ij (i, j, � 1, 2, · · · , n), and the product is " ∝ " . erefore, us, logical equations (19) are solved if and only if there that is, the v-th column of M is zero. en, the proof is complete. □ Based on eorem 2, we give the following algorithm to find all for the L(p, q) coloring solutions of the given graph.

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 T-coloring, list coloring, set coloring, and L(p 1 , p 2 , · · · , p n )-coloring.
e most commonly used model is L(2, 1)-coloring. Denote by G(V, E) the topological graph of the wireless network, where V is a node set denoting base stations or their users and E denotes an edge set. Now, we will use the L(2, 1)-coloring model to analyze the channel allocation of the wireless network. Example 1. Construct the telecommunication base stations among four cities denoted by A ∼ D, respectively. Denote a city by one vertex of the graph. If the base station constructed in city X can cover city Y, then there is an edge between X and Y. Now, the covering graph of base stations G � V, E { } 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 A, we have by the above definition B � 0 1 1 1 1 0 1 1 Using Algorithm 1, we can compute the matrix M defined in (26) by the semitensor product. en, by Matlab, k � 5 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 to   Table 1. For example, for i � 97, y 1 � 4, y 2 � 1, y 3 � 0, y 4 � 3. at 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.

Conclusion
e L(p, q)-coloring problem of the graph is studied with application to channel allocation of the wireless network in this paper. e necessary and sufficient conditions for solving the L(p, q)-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., T-coloring, list coloring, and set coloring.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.