Graph Coloring Theory Research Articles

Application of Multiobjective Optimization Integrating Numerical and Scientific Computing in Graph Theory Coloring Algorithm

In the current era of increasingly complex social life, as people’s demand quality is getting higher and higher, the solution of the problem often has multiple indicators to achieve the optimum. This forces the graph coloring problem (GCP) to become complicated, and it is difficult to directly obtain the optimal solution, which brings new challenges to the solution of the problem. In response to this problem, the GCP field is of great research significance. With the in-depth study of GCP, the research on multiobjective optimization (MOO) in graph coloring algorithm (GCA) is gradually carried out. Its performance advantage is of great significance for solving multicondition constraint problems. This paper aims to study the application of Genetic Algorithm (GA) in GCA. Through the analysis and research of GA, and the fusion of numerical analysis and scientific computing, it can be applied to the construction of the neighbor distinguishable uniform V-full coloring algorithm (AVDEVTCA) to solve the AVDEVTC problem. This paper describes the basic theory of MOO and graph coloring. It conducts an experimental analysis of the algorithm performance and uses the relevant theoretical formulas to explain it. The results show that the algorithm takes 5754.142 S seconds to test 21325415 images and can color a large number of results that cannot be done manually in less time. It has greatly improved in terms of time and manpower saving and greatly improved practicability and work efficiency.

Resource Allocation for D2D Communication in Cellular Networks Based on Stochastic Geometry and Graph-coloring Theory

In a device-to-device (D2D) underlaid cellular network, there exist two types of co-channel interference. One type is inter-layer interference caused by spectrum reuse between D2D transmitters and cellular users (CUEs). Another type is intra-layer interference caused by spectrum sharing among D2D pairs. To mitigate the inter-layer interference, we first derive the interference limited area (ILA) to protect the coverage probability of cellular users by modeling D2D users’ location as a Poisson point process, where a D2D transmitter is allowed to reuse the spectrum of the CUE only if the D2D transmitter is outside the ILA of the CUE. To coordinate the intra-layer interference, the spectrum sharing criterion of D2D pairs is derived based on the (signal-to-interference ratio) SIR requirement of D2D communication. Based on this criterion, D2D pairs are allowed to share the spectrum when one D2D pair is far from another sufficiently. Furthermore, to maximize the energy efficiency of the system, a resource allocation scheme is proposed according to weighted graph coloring theory and the proposed ILA restriction. Simulation results show that our proposed scheme provides significant performance gains over the conventional scheme and the random allocation scheme.

A Fast Calibration Method for Phased Arrays by Using the Graph Coloring Theory.

Phased array radars are able to provide highly accurate airplane surveillance and tracking performance if they are properly calibrated. However, the ambient temperature variation and device aging could greatly deteriorate their performance. Currently, performing a calibration over a large-scale phased array with thousands of antennas is time-consuming. To facilitate the process, we propose a fast calibration method for phased arrays with omnidirectional radiation patterns based on the graph coloring theory. This method transforms the calibration problem into a coloring problem that aims at minimizing the number of used colors. By reusing the calibration time slots spatially, more than one omnidirectional antenna can perform calibration simultaneously. The simulation proves this method can prominently reduce total calibration time and recover the radiation pattern from amplitude and phase errors and noise. It is worth noting that the total calibration time consumed by the proposed method remains constant and is negligible compared with other calibration methods.

On certain coloring parameters of Mycielski graphs of some graphs

Coloring the vertices of a graph [Formula: see text] according to certain conditions can be considered as a random experiment and a discrete random variable [Formula: see text] can be defined as the number of vertices having a particular color in the proper coloring of [Formula: see text]. The concepts of mean and variance, two important statistical measures, have also been introduced to the theory of graph coloring and determined the values of these parameters for a number of standard graphs. In this paper, we discuss the coloring parameters of the Mycielskian of certain standard graphs.

Types of fuzzy graph coloring and polynomial ideal theory

Types of fuzzy graph coloring and polynomial ideal theory

On certain parameters of equitable coloring of graphs

Coloring the vertices of a graph [Formula: see text] according to certain conditions is a random experiment and a discrete random variable [Formula: see text] is defined as the number of vertices having a particular color in the given type of coloring of [Formula: see text] and a probability mass function for this random variable can be defined accordingly. An equitable coloring of a graph [Formula: see text] is a proper coloring [Formula: see text] of [Formula: see text] which an assignment of colors to the vertices of [Formula: see text] such that the numbers of vertices in any two color classes differ by at most one. In this paper, we extend the concepts of arithmetic mean and variance, the two major statistical parameters, to the theory of equitable graph coloring and hence determine the values of these parameters for a number of standard graphs.

An uncertain chromatic number of an uncertain graph based on $$\alpha $$ α -cut coloring

An uncertain graph is a graph in which the edges are indeterminate and the existence of edges are characterized by belief degrees which are uncertain measures. This paper aims to bring graph coloring and uncertainty theory together. A new approach for uncertain graph coloring based on an $$\alpha $$ -cut of an uncertain graph is introduced in this paper. Firstly, the concept of $$\alpha $$ -cut of uncertain graph is given and some of its properties are explored. By means of $$\alpha $$ -cut coloring, we get an $$\alpha $$ -cut chromatic number and examine some of its properties as well. Then, a fact that every $$\alpha $$ -cut chromatic number may be a chromatic number of an uncertain graph is obtained, and the concept of uncertain chromatic set is introduced. In addition, an uncertain chromatic algorithm is constructed. Finally, a real-life decision making problem is given to illustrate the application of the uncertain chromatic set and the effectiveness of the uncertain chromatic algorithm.

A unified proof of Brooks’ theorem and Catlin’s theorem

Brooks’ theorem is a fundamental result in the theory of graph coloring. Catlin proved the following strengthening of Brooks’ theorem: Let d be an integer at least 3, and let G be a graph with maximum degree d. If G does not contain Kd+1 as a subgraph, then G has a d-coloring in which one color class has size α(G). Here α(G) denotes the independence number of G. We give a unified proof of Brooks’ theorem and Catlin’s theorem.

Joint Resources Assignment Scheme Based on Graph Coloring Theory and Harmony Search Theory for Cognitive Radio Networks

Spectrum assignment and power assignment are both key technology to use spectrum resource for cognitive radio networks effectively. In this paper, an effective joint resources assignment method is proposed. The objective is to maximize the system throughput and to guarantee the quality of service for the users simultaneously. The method is based on harmony search algorithm and graph coloring theory. It gives a more proper representation of the interference between the users in our system model. Thus, the users can reuse the spectrum more effectively. Simulations show that the proposed method can achieve higher system throughput in the new model than that in the traditional graph model.

An Effective Spectrum Allocation Method Based on Color Sensitive Graph Coloring Theory for CR Networks

Spectrum allocation is one of the key technologies of cognitive radio networks. The purpose of spectrum allocation is to maximize the utilization efficiency of available spectrum resources and to ensure user fairness by choosing and using the “free spectrum” reasonably. In this paper, an effective spectrum allocation method based on improved color sensitive graph coloring theory is presented. The proposed method gives a more proper representation of the interference between the users. Thus, the users can reuse the spectrum more effectively. Meanwhile, the proposed algorithm considers the demands of different users when distributing spectrum resources to ensure the fairness. Simulation results testify the effectiveness of the proposed algorithm. It performs better than existing methods in the aspects of spectrum utilization efficiency and user fairness.

Toeplitz Compressed Sensing Matrices With Applications to Sparse Channel Estimation

Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. In essence, CS enables the recovery of high-dimensional sparse signals from relatively few linear observations in the form of projections onto a collection of test vectors. Existing results show that if the entries of the test vectors are independent realizations of certain zero-mean random variables, then with high probability the unknown signals can be recovered by solving a tractable convex optimization. This work extends CS theory to settings where the entries of the test vectors exhibit structured statistical dependencies. It follows that CS can be effectively utilized in linear, time-invariant system identification problems provided the impulse response of the system is (approximately or exactly) sparse. An immediate application is in wireless multipath channel estimation. It is shown here that time-domain probing of a multipath channel with a random binary sequence, along with utilization of CS reconstruction techniques, can provide significant improvements in estimation accuracy compared to traditional least-squares based linear channel estimation strategies. Abstract extensions of the main results are also discussed, where the theory of equitable graph coloring is employed to establish the utility of CS in settings where the test vectors exhibit more general statistical dependencies.

Some Topological Methods in Graph Coloring Theory

Abstract Attempts to solve the famous Four Color Problem led to fruitful discoveries and rich coloring theories. In this talk, some old and some recent applications of tools from topology to graph coloring problems will be presented. In particular, the following subjects will be treated: The use of Euler's formula and local planarity conditions, Kempe equivalence, homotopy, winding number and its higher dimensional analogues.

PLADE: a two-stage PLA decomposition

An efficient method for PLA decomposition, in which a single two-level Boolean function (PLA) is decomposed into two stages of cascaded PLAs such that the total area of all PLAs is smaller than that of the original PLA, is presented. The first stage may contain an arbitrary number of PLAs (generalized decoders), and the second stage contains a single PLA. Primary inputs are partitioned into disjoint sets of input variables and represented as multiple-valued variables so as to minimize the total PLA area. Two efficient algorithms for assignment of input variables to individual decoders are presented, one based on an integer programming technique and the other on graph partitioning. The constrained input encoding of the second-stage PLA is performed by an efficient procedure based on graph coloring and partitioning theory.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>