‎Graph Coloring is the assignment of one color to each vertex of a Graph so that two adjacent vertices are not of the same color‎. ‎The Graph Coloring problem (GCP) is a matter of combinatorial optimization‎, ‎and the goal of GCP is determining the chromatic number $\chi(G)$‎. ‎Since GCP is an NP-hard problem‎, ‎then in this paper‎, ‎we propose a new approximated algorithm for finding the Coloring number (it is an approximation of chromatic number) by using a Graph adjacency matrix to colorize or separate a Graph‎. ‎To prove the correctness of the proposed algorithm‎, ‎we implement it in MATLAB software‎, ‎and for analysis in terms of solution and execution time‎, ‎we compare our algorithm with some of the best existing algorithms that are already implemented in MATLAB software‎, ‎and we present the results in tables of various Graphs‎. ‎Several available algorithms used the largest degree selection strategy‎, ‎while our proposed algorithm uses the Graph adjacency matrix to select the vertex that has the smallest degree for Coloring‎. ‎We provide some examples to compare the performance of our algorithm to other available methods‎. ‎We make use of the Dolan-Mor\'e performance profiles to assess the performance of the numerical algorithms‎, ‎and demonstrate the efficiency of our proposed approach in comparison with some existing methods‎.

Web services as independent software components are published on the Internet by service providers and services are then called by users’ request. However, in many cases, no service alone can be found in the service repository that could satisfy the applicant satisfaction. Service composition provides new components by using an interactive model to accelerate the programs. Prior to service composition, the most important issue in finding suitable candidate services samples is their compliance with non-functional requirements. Thus, designing an efficient way to combine a chain of connected services is important. Recently, numerous studies have been done to reduce the search time in finding a service composition. However, many of these methods to examine and investigate all Web services in a Web repository require a long time, which occupy the user's time significantly. This paper provides an approach for automatic quality-aware service composition as well as the users’ preferences in achieving the optimum composition results. For this purpose, modified Graph Coloring method to filter the data before compositions in large-scale data is used which decreases selected services set. The application of KPL algorithm in this study provided some proper solutions to the user so that these solutions can be used instead of the best composition if necessary. Therefore, the results derived from the analysis of the proposed method, indicates a good optimization in runtime and memory consumption. The evaluation results show that the proposed method in memory consumption and runtime has improved by about 20%.

Graph Coloring is one of the issues that has been most noticed among combinatorial optimization issues. Many useful utility issues can be modeled as Graph Coloring issues. The general form of this application is to form a Graph with nodes representing our favorite parts. The main problem of Coloring the Graph is the grouping of vertex Graphs in small groups, so that no two heterogeneous vertices are in the same group. An important part of the application of Graph Coloring problem in management science is. The concept of traffic lights includes controlling the system of a traffic light so that a safe level of safety can be obtained. Modeling the problem of traffic lights has been proposed as a problem of assignment in combinatorial theory. This problem is also modeled as a Graph Coloring problem. In this paper, we have tried to model these problems in practical examples as the problem of staining the fuzzy Graph and compare them with the proposed methods.

This paper begins with a review of the Graph Colouring Problem and its literature. Next developed methods for solving the problem are examined. Intelligent methods have demonstrated superior efficiency, since the problem is a NP-hard benchmark. A co-evolutionary algorithm for solving this problem is presented in this paper. The algorithm consists of two populations that simultaneously evolve using genetic operations. Eventually, the complete solution is found through merging individual members of the two populations. The effectiveness of the procedure has been compared to that of a simple genetic algorithm. The obtained results show significant advantage over the traditional routines.

In this paper, we find the star chromatic number of central Graph of complete bipartite Graph and corona Graph of complete Graph with path and cycle.

In this paper we investigate the dominating--color number، of a Graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and H. This result allows us to construct classes of Graphs such that and thus provide some information regarding two questions raised in [1] and [2].

It is important to have an accurate and reliable brain tumor segmentation for cancer diagnosis and treatment planning. There are few unsupervised approaches for brain tumor segmentation. In this paper, a new unsupervised approach based on Graph Coloring for brain tumor segmentation is introduced. In this study, a Graph Coloring approach is used for brain tumor segmentation. For this aim, each pixel of brain image assumed as a node of Graph and difference between brightness of a couple of pixels considered as edge. This method was applied on T1-enhanced magnetic resonance images of low-grade and high-grade patients. Since a rigid Graph was needed for Graph Coloring, edges must be divided into existing or nonexisting edge using a threshold. The value of this threshold has affected the accuracy of image segmentation, so the choice of the optimal threshold was important. The optimal value for this threshold was 0. 42 of maximum value of difference of brightness between pixels that caused the 83. 62% of correlation accuracy. The results showed that Graph Coloring approach can be a reliable unsupervised approach for brain tumor segmentation. This approach, as an unsupervised approach, shows better accuracy in comparison with neural networks and neuro-fuzzy networks. However, as a limitation, the accuracy of this approach is dependent on the threshold of edges.

Massive MIMO cellular networks, despite their ability to serve multiple users simultaneously, face a significant challenge due to pilot contamination. This paper presents an innovative two-stage algorithm to reduce this contamination and increase user data rates in both uplink and downlink. The key innovation of the proposed method lies in the intelligent integration of three techniques: Soft Pilot Reuse (SPR), optimal pilot sequence selection algorithm, and Weighted Graph Coloring (WGC). This combination simultaneously addresses three fundamental issues: contamination in SPR due to fixed thresholds, increased pilot overhead in WGC, and the problem of maximizing the data rate of the user with the lowest rate. The proposed algorithm operates in two stages. In the first stage, users are divided into center and edge groups, and optimal pilot sequences are determined based on their data rates. In the second stage, using the WGC algorithm and creating an Edge-Weighted Interference Graph (EWIG), pilot contamination is reduced based on the intensity of user interference. Simulation results show that this method significantly improves system performance compared to the best existing method (WGC). In the downlink, an 11 dB improvement in Signal-to-Interference-plus-Noise Ratio (SINR) and a 0.16 bps/Hz increase in average achievable rate are observed. In the uplink, a 2.4 dB improvement in SINR and a 0.46 bps/Hz increase in average achievable rate are achieved. In terms of computational complexity, the proposed method has lower complexity compared to the WGC scheme. Moreover, energy efficiency analysis confirms the superiority of the proposed method in both uplink and downlink scenarios.

With regard to roads and traffic routes to different levels of risk, it can be determined which of these sections of these roads and paths are more likely to crash and based on this risk, the maximum speed and the various authorized routes As low as possible. In this study, using a fuzzy Graph as a math model of the urban chip network, we present a method for determining the different areas of traffic in terms of the level of risk. Based on the probability of accidents, traffic areas fall into three low risk areas (green), in the danger zone (yellow) and high risk (red). In this study, a Graph Coloring method is presented that includes two automatic solitary and fuzzy logic segments. In this method, we perform the Coloring of the existing Graph using a fuzzy automaton system. In this study, the amount of α,indicates the quality sensitivity of that road and it has been shown that increasing the amount of α,will increase the number of most dangerous routes and increase the sensitivity to driving quality.