New setting is introduced to study resolving number and chromatic number alongside dominating number. Different types of procedures including set, optimal set, and optimal number alongside study on the family of neutrosophic hypergraphs are proposed in this way, some results are obtained. General classes of neutrosophic hypergraphs are used to obtains these numbers and the representatives of the colors, dominating sets and resolving sets. Using colors to assign to the vertices of neutrosophic hypergraphs and characterizing resolving sets and dominating sets are applied. Some questions and problems are posed concerning ways to do further studies on this topic. Using different ways of study on neutrosophic hypergraphs to get new results about numbers and sets in the way that some numbers get understandable perspective. Family of neutrosophic hypergraphs are studied to investigate about the notions, dimension and coloring alongside domination in neutrosophic hypergraphs. In this way, sets of representatives of colors, resolving sets and dominating sets have key role. Optimal sets and optimal numbers have key points to get new results but in some cases, there are usages of sets and numbers instead of optimal ones. Simultaneously, three notions are applied into neutrosophic hypergraphs to get sensible results about their structures. Basic familiarities with neutrosophic hypergraphs theory and hypergraph theory are proposed for this article.

New setting is introduced to study types of coloring numbers, degree of vertices, degree of hyperedges, co-degree of vertices, co-degree of hyperedges, neutrosophic degree of vertices, neutrosophic degree of hyperedges, neutrosophic co-degree of vertices, neutrosophic co-degree of hyperedges, neutrosophic number of vertices, neutrosophic number of hyperedges in neutrosophic hypergraphs. Different types of procedures including neutrosophic (r, n)−regular hypergraphs and neutrosophic complete r−partite hypergraphs are proposed in this way, some results are obtained. General classes of neutrosophic hypergraphs are used to obtain chromatic number, the representatives of the colors, degree of vertices, degree of hyperedges, co-degree of vertices, co-degree of hyperedges, neutrosophic degree of vertices, neutrosophic degree of hyperedges, neutrosophic co-degree of vertices, neutrosophic co-degree of hyperedges, neutrosophic number of vertices, neutrosophic number of hyperedges in neutrosophic hypergraphs. Using colors to assign to the vertices of neutrosophic hypergraphs and characterizing representatives of the colors are applied in neutrosophic (r, n)−regular hypergraphs and neutrosophic complete r−partite hypergraphs. Some questions and problems are posed concerning ways to do further studies on this topic. Using different ways of study on neutrosophic hypergraphs to get new results about number, degree and co-degree in the way that some number, degree and co-degree get understandable perspective. Neutrosophic (r, n)−regular hypergraphs and neutrosophic complete r−partite hypergraphs are studied to investigate about the notions, coloring, the representatives of the colors, degree of vertices, degree of hyperedges, co-degree of vertices, co-degree of hyperedges, neutrosophic degree of vertices, neutrosophic degree of hyperedges, neutrosophic co-degree of vertices, neutrosophic co-degree of hyperedges, neutrosophic number of vertices, neutrosophic number of hyperedges in neutrosophic (r, n)−regular hypergraphs and neutrosophic complete r−partite hypergraphs. In this way, sets of representatives of colors, degree of vertices, degree of hyperedges, co-degree of vertices, co-degree of hyperedges, neutrosophic degree of vertices, neutrosophic degree of hyperedges, neutrosophic co-degree of vertices, neutrosophic co-degree of hyperedges, neutrosophic number of vertices, neutrosophic number of hyperedges have key points to get new results but in some cases, there are usages of sets and numbers instead of optimal ones. Simultaneously, notions chromatic number, the representatives of the colors, degree of vertices, degree of hyperedges, co-degree of vertices, co-degree of hyperedges, neutrosophic degree of vertices, neutrosophic degree of hyperedges, neutrosophic co-degree of vertices, neutrosophic co-degree of hyperedges, neutrosophic number of vertices, neutrosophic number of hyperedges are applied into neutrosophic hypergraphs, especially, neutrosophic (r, n)−regular hypergraphs and neutrosophic complete r−partite hypergraphs to get sensible results about their structures. Basic familiarities with neutrosophic hypergraphs theory and hypergraph theory are proposed for this article.

New setting is introduced to study “closing numbers” and “super-closing numbers” as optimal-super-resolving number, optimal-super-coloring number and optimal-super-dominating number. In this way, some approaches are applied to get some sets from (Neutrosophic)n-SuperHyperGraph and after that, some ideas are applied to get different types of super-closing numbers which are called by optimal-super-resolving number, optimal-super-coloring number and optimal-super-dominating number. The notion of dual is another new idea which is covered by these notions and results. In the setting of dual, the set of super-vertices is exchanged with the set of super-edges. Thus these results and definitions hold in the setting of dual. Setting of neutrosophic n-SuperHyperGraph is used to get some examples and solutions for two applications which are proposed. Both setting of SuperHyperGraph and neutrosophic n-SuperHyperGraph are simultaneously studied but the results are about the setting of n-SuperHyperGraphs. Setting of neutrosophic n-SuperHyperGraph get some examples where neutrosophic hypergraphs as special case of neutrosophic n-SuperHyperGraph are used. The clarifications use neutrosophic n-SuperHyperGraph and theoretical study is to use n-SuperHyperGraph but these results are also applicable into neutrosophic n-SuperHyperGraph. Special usage from different attributes of neutrosophic n-SuperHyperGraph are appropriate to have open ways to pursue this study. Different types of procedures including optimal-super-set, and optimal-super-number alongside study on the family of (neutrosophic)n-SuperHyperGraph are proposed in this way, some results are obtained. General classes of (neutrosophic)n-SuperHyperGraph are used to obtains these closing numbers and super-closing numbers and the representatives of the optimal-super-coloring sets, optimal-super-dominating sets and optimal-super-resolving sets. Using colors to assign to the super-vertices of n-SuperHyperGraph and characterizing optimal-super-resolving sets and optimal-super-dominating sets are applied. Some questions and problems are posed concerning ways to do further studies on this topic. Using different ways of study on n-SuperHyperGraph to get new results about closing numbers and super-closing numbers alongside sets in the way that some closing numbers super-closing numbers get understandable perspective. Family of n-SuperHyperGraph are studied to investigate about the notions, super-resolving and super-coloring alongside super-dominating in n-SuperHyperGraph. In this way, sets of representatives of optimal-super-colors, optimal-super-resolving sets and optimal-super-dominating sets have key role. Optimal-super sets and optimal-super numbers have key points to get new results but in some cases, there are usages of sets and numbers instead of optimal-super ones. Simultaneously, three notions are applied into (neutrosophic)n-SuperHyperGraph to get sensible results about their structures. Basic familiarities with n-SuperHyperGraph theory and neutrosophic n-SuperHyperGraph theory are proposed for this article.

For a positive integer k, a radio k-coloring of a simple connected graph G = (V (G), E(G)) is a mapping | f(u) - f(v)| ≥ k +1-d (u , v ) such that f :V (G)→{0,1, 2,...} for each pair of distinct vertices u and v of G, where d(u, v) is the distance between u and v in G. The span of a radio k-coloring f, rck(f), is the maximum integer it assigns to some vertex of G. The radio k-chromatic number, rck(G) of G is min{rck(f)}, where the minimum is taken over all radio k-colorings f of G. If k is the diameter of G, then rck(G) is known as the radio number of G. In this work, we propose four algorithms (two serial algorithms and their parallel versions) which related to the radio k-coloring problem. One of them is an approximate algorithm that determines an upper bound of the radio number of a given graph. The other is an exact algorithm which finds the radio number of a graph G. The approximate algorithm is a polynomial time algorithm while the exact algorithm is an exponential time algorithm. The parallel algorithms are parallelized using the Message Passing Interface (MPI) standard. The experimental results prove the ability of the proposed algorithms to achieve a speedup 7 for 8 processors.

Existing proofs of the 4-color theorem succeeded by establishing an unavoidable set of reducible configurations. By this device, their authors showed that a minimum counterexample cannot exist. G.D. Birkhoff proved that a minimum counterexample must satisfy a connectivity property that is referred to in modern parlance as internal 6-connectivity. We show that a minimum counterexample must also satisfy a coloring property, one that we call Kempe-locking. We define the terms Kempe-locking configuration and fundamental Kempe-locking configuration. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked; it involves deconstructing the triangulation into a stack of configurations with common endpoints and then creating a bipartite graph of coloring possibilities for each configuration in the stack to assess whether certain 2-color paths can be transmitted from the configuration's top boundary to its bottom boundary. All Kempe-locked triangulations we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say $xy$, and (2) they have a Birkhoff diamond with endpoints $x$ and $y$ as a proper subgraph. On the strength of our various investigations, we are led to a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample to the 4-color theorem are incompatible. It would also point to the singular importance of a particularly elegant 4-connected triangulation of order 9 that consists of a triangle enclosing a pentagon enclosing a single vertex.

Jacobian-free Newton Krylov (JFNK) is an attractive method to solve nonlinear equations in nuclear engineering systems, and has been successfully applied to steady-state neutron diffusion (k-eigenvalue) problems and multi-physics coupling problems. Preconditioning technique plays an important role in JFNK algorithm, significantly affecting computational efficiency. The key point is how to automatically construct a high-quality preconditioning matrix that can improve the convergence rate, and perform the preconditioning matrix factorization efficiently. An efficient reordering-based ILU(k) preconditioner is proposed to achieve the above objectives. In detail, the finite difference technique combined with the coloring algorithm is utilized to construct an efficient preconditioning matrix with low computational cost automatically. Furthermore, the reordering algorithm is employed for the ILU(k) to reduce the additional non-zero elements and pursue robust computational performance. A two-dimensional LRA neutron steady-state benchmark problem is used to evaluate the performance of the proposed preconditioning technique, and steady-state neutron diffusion problem with thermal-hydraulic feedback is also utilized as a supplement. The results show that coloring algorithms can efficiently and automatically construct the preconditioning matrix. The reordering-based ILU(k) preconditioner shows excellent robustness, avoiding the effect of fill-in levels k choice in incomplete LU factorization.

The idea of strong and weak adjacencies between vertices has been generalized into fuzzy graphs and intuitionistic fuzzy graphs (IFGs) and it is an important part of making decisions. However, one or two membership degrees are not always sufficient for making decisions on real-world problems that need an answer of types “yes, neutral, and no". Consequently, in previous work, we generalized the concept into picture fuzzy graphs (PFGs) where each element in the PFG has membership, neutral, and non-membership degrees. Moreover, we constructed the notion of the coloring of PFGs based on strong and weak adjacencies between vertices. In this paper, we investigate some properties of the chromatic number of PFGs based on the concept of strong and weak adjacencies between vertices. According to the properties, we construct an algorithm to find the chromatic number of PFGs. The algorithm is usefull when we work with large PFGs. Further, we improve the method to implement the PFG’s coloring for determining traffic signal phasing at an intersection. A case study has also been done to evaluate the method.

The Loropetalum chinense and Loropetalum chinense var. rubrum are typical and traditional ornamental and Chinese herbal medicine in Asia. However, more information is needed on the mechanisms underlying its flower coloring. Here, we profiled the flavonoid metabolome, full-length sequencing, and transcriptome analysis to investigate the flavonoid biosynthesis and global transcriptome changes among different petal coloring cultivars of L. chinense and L. chinense var. rubrum. The total anthocyanins and phenotypic of the petal were highly consistent with the petal color. Moreover, a total of 207 flavonoid components were identified. Of these, 12 flavonoid components were considered significantly different expression compounds among the four samples. Meanwhile, the first reference full-length transcriptome of L. chinense var. rubrum was being built, which had 171,783 high-quality non-redundant transcripts with correcting with next-generation sequencing (NGS). Among them, 52,851 transcripts were annotated in the seven database of NR, KOG, GO, NT, Pfam, Swiss-Port and KEGG. Combined with NGS analysis, the DETs involved in flavonoids and anthocyanins contributed greatest to the flower-coloring. Additionally, the different expressed of eight LcDFRs and four LcANS genes were positively correlated with flavonoid biosynthesis, and the four LcBZ1 and one Lc3Mat1 were positively correlated with the content of seven anthocyanins revealed by coupling with metabolomics and transcriptomics analysis. Together these results were used to mine candidate genes by analyzing flower coloring changes in a comprehensive metabolic and transcriptomic level in L. chinense and L. chinense var. rubrum.

Visual thinking leverages spatial mechanisms in animals for navigation and reasoning. Therefore, given the challenge of abstract mathematics and logic, spatial reasoning-based teaching strategies can be highly effective. Our previous research verified that innovative box and ball coloring activities help teach elementary school students complex notions like quantifiers, logical connectors, and dynamic systems. However, given the richness of the activities, correction is slow, error-prone, and demands high attention and cognitive load from the teacher. Moreover, feedback to the teacher should be immediate. Thus, we propose to provide the teacher with real-time help with LLMs. We explored various prompting techniques with and without context Zero shot, Few shot, Chain of Thought, Visualization of Thought, Self Consistency, logicLM and Emotional to test GPT-4o’s visual, logical, and correction capabilities. We obtained that Visualization of Thought and Self Consistency techniques enabled GPT-4o to correctly evaluate 90% of logical-spatial problems that we tested. Additionally, we propose a novel prompt combining some of these techniques that achieved 100% accuracy on a testing sample, excelling in spatial problems and enhancing logical reasoning.

With the continuous development of 3D city modeling, traditional close-range photogrammetry is limited by complex processing procedures and incomplete 3D depth information, making it unable to meet high-precision modeling requirements. In contrast, the integration of LiDAR and camera in mobile measurement systems provides a new and highly effective solution. The LiDAR can quickly and accurately acquire the 3D spatial coordinates of target objects, while optical imagery contains abundant color information. If the two can be integrated, they can play an important role in multiple fields such as streetscape modeling, archaeology and digital city, etc. Currently, integrated mobile measurement systems commonly require cameras, lasers, POS and IMU, thus the hardware cost is relatively expensive and the system integration is complex. Therefore, in this paper we propose a simple ground mobile measurement system composed of a LiDAR and a GoPro camera without a POS system, providing a more convenient and reliable way to automatically obtain 3D point cloud data with spectral information. The automatic point cloud coloring based on video images mainly includes four aspects: (1) Establishing models for radial distortion and tangential distortion to correct video images. (2) Establishing a registration method based on normalized Zernike moments to obtain the exterior orientation elements. Normalized Zernike moments are region-based shape descriptors that can reflect the features of images in multiple dimensions even for low-quality video images. The results show that registration based on normalized Zernike moments provides a good result, with an error accuracy of 0.5-1 pixel, which is far higher than registration based on a collinearity equation. (3) Establishing adjacent video image relative orientation based on essential matrix decomposition and nonlinear optimization. This involves uniformly using the SURF algorithm with distance restriction and RANSAC to select corresponding points, which can improve the reliability of the corresponding points. The results indicate that the accuracy of the relative orientation method is high. Moreover, this method can converge to good results for stereo image pairs with large rotation angles and displacement amounts. Therefore, relative orientation based on essential matrix decomposition and nonlinear optimization has good applicability. (4) The video imagery suffers from significant motion blur and boundary distortion. Therefore, a point cloud coloring method based on Gaussian distribution with central region restriction is adopted. Only pixels within the central region are considered valid for coloring. Then, the point cloud is colored based on the mean of the Gaussian distribution of the color set. Experimental results show that the coloring accuracy between the video imagery and point cloud data is high, meeting the accuracy requirements of applications such as tunnel detection, street-view modeling and 3D urban modeling.

This paper offers a novel geometric and visual approach to the renowned Four Color Map Theorem and K5 non-planarity problem while unveiling their profound connection to the "kissing number" problem in 2D. We represent planar graphs through circles and tangents, simplifying complex structures and shedding light on these classic problems. Our proofs by contradiction, rooted in the kissing number concept, reveal that both the Four Color Map Theorem and K5 non-planarity are fundamentally linked, as they pivot around the concept of coloring. This study bridges the realms of geometry and graph theory, providing fresh insights and emphasizing the significance of the kissing number problem in various fields.