Coloring Problem Research Articles

Parameterized complexity for iterated type partitions and modular-width

This paper deals with the complexity of some natural graph problems parameterized by some measures that are restrictions of clique-width, such as modular-width and neighborhood diversity. We introduce a novel parameter, called iterated type partition number, that can be computed in linear time and nicely places between modular-width and neighborhood diversity. We prove that the Equitable Coloring problem is W[1]-hard when parameterized by the iterated type partition number. This result extends to modular-width, answering an open question on the complexity of Equitable Coloring when parameterized by modular-width. On the contrary, we show that the Equitable Coloring problem is FPT when parameterized by neighborhood diversity.Furthermore, we present a scheme for devising FPT algorithms parameterized by iterated type partition number, which enables us to find optimal solutions for several graph problems. As an example, in this paper, we present algorithms parameterized by the iterated type partition number of the input graph for some generalized versions of the Maximum Clique, Minimum Graph Coloring, (Total) Minimum Dominating Set, Minimum Vertex Cover and Maximum Independent Set problems. Each algorithm outputs not only the optimal value but also the optimal solution. We stress that while the considered problems are already known to be FPT with respect to modular-width, the novel algorithms are both simpler and more efficient. We finally show that the proposed scheme can be used to devise polynomial kernels, with respect to iterated type partition number, for the decisional version of most of the problems mentioned above.

A Solution to Graph Coloring Problem Using Genetic Algorithm

INTRODUCTION: The Graph Coloring Problem (GCP) involves coloring the vertices of a graph in such a way that no two adjacent vertices share the same color while using the minimum number of colors possible.
 OBJECTIVES: The main objective of the study is While keeping the constraint that no two neighbouring vertices have the same colour, the goal is to reduce the number of colours needed to colour a graph's vertices. It further investigate how various techniques impact the execution time as the number of nodes in the graph increases.
 METHODS: In this paper, we propose a novel method of implementing a Genetic Algorithm (GA) to address the GCP.
 RESULTS: When the solution is implemented on a highly specified Google Cloud instance, we likewise see a significant increase in performance. The parallel execution on Google Cloud shows significantly faster execution times than both the serial implementation and the parallel execution on a local workstation. This exemplifies the benefits of cloud computing for computational heavy jobs like GCP.
 CONCLUSION: This study illustrates that a promising solution to the Graph Coloring Problem is provided by Genetic Algorithms. Although the GA-based approach does not provide an optimal result, it frequently produces excellent approximations in a reasonable length of time for a variety of real-world situations.

Universality of Graph Homomorphism Games and the Quantum Coloring Problem

Universality of Graph Homomorphism Games and the Quantum Coloring Problem

Formula omitted]-packing coloring of cubic Halin graphs

Given a non-decreasing sequence S=(s1,s2,…,sk) of positive integers, an S-packing coloring of a graph G is a partition of the vertex set of G into k subsets {V1,V2,…,Vk} such that for each 1≤i≤k, the distance between any two distinct vertices u and v in Vi is at least si+1. In this paper, we study the problem of S-packing coloring of cubic Halin graphs, and we prove that every cubic Halin graph is (1,1,2,3)-packing colorable. In addition, we prove that such graphs are (1,2,2,2,2,2)-packing colorable.

Intersectionality of Slavery and Gender Analyzing the Experiences of Black Women in Selected Writings

Intersectionality has created a new caste system which strictly adheres to the problem of colour, segregation, injustice, violence, and categorization on the basis of social class, race, and gender. The interconnected nature of social categorization combines multiple identities including racial identity, sexuality, disability, nationality, and gender; intersects each other, and results into discrimination and oppression. All forms of racial and gender inequalities reinforce each other, fortify identity politics, and ostracize on the basis of gender, caste and colour of skin. Slavery as an institution has also colossally impacted the question of gender inequality in American society. In the epoch of slavery, the women of different colour were the major victims of two-fold exploitation as being a woman and being a black racially. They were prime targets of sexual exploitation, physical violence, and free labor. In African American Literature, Toni Morrison’s Beloved and Harriet Jacobs’ Incidents in the Life of a Slave Girl are the two most prominent writings which depict the adverse effects of slavery, which has ruined several generations of the Blacks. The experiences of black women as enslaved workers, and victims of sexual exploitation are not less than a nightmare. The selected writings explicitly expose the ugly face of slavery that kept the black people particularly women in iron-made baskets for centuries. The study focuses on the discourse of two selected texts that how were black women maltreated, marginalized, sexually exploited and dehumanized.

The Development of RBL-STEM Learning Materials to Improve Students' Information Literacy in Solving Rainbow Antimagic Coloring Problem for ETLE Technology

Students often struggle to solve complex mathematical problems in real-world contexts due to low information literacy skills. To improve information literacy skills, an effective learning approach can be RBL-STEM, which provides research-based learning and can be practically applied in the real world. This study aims to investigate RBL-STEM activities, describe the process and results of developing RBL-STEM materials, and analyze data from the results of developing these materials to improve students’ information literacy skills. The method of research used is research and development (R&D). The purpose of this research is to develop RBL-STEM materials and produce learning material products in the form of semester learning plans, student assignment designs, student worksheets, and learning outcomes tests. The results of the materials development showed validity with a validity criterion of 92.75%. The trial was conducted with 40 students, and the implementation using the RBL-STEM materials was found to be practical with a practical criterion of 98.75% and effective with an effectiveness criterion of 94%. In addition, the students were highly engaged and responded very positively to the learning experience. Pretest and posttest analysis showed an increase in students’ information literacy in solving the rainbow antimagic coloring problem. The study also identified three levels of information literacy skills: high, medium, and low. Statistical analysis, phase portrait, NVivo, and word cloud confirmed the research findings and showed an increase in students’ information literacy skills. Thus, RBL-STEM has the potential to improve students’ information literacy in real-world contexts, such as the application of ETLE using graph neural network techniques.

Improving Text-Based Person Retrieval by Excavating All-Round Information Beyond Color.

Text-based person retrieval is the process of searching a massive visual resource library for images of a particular pedestrian, based on a textual query. Existing approaches often suffer from a problem of color (CLR) over-reliance, which can result in a suboptimal person retrieval performance by distracting the model from other important visual cues such as texture and structure information. To handle this problem, we propose a novel framework to Excavate All-round Information Beyond Color for the task of text-based person retrieval, which is therefore termed EAIBC. The EAIBC architecture includes four branches, namely an RGB branch, a grayscale (GRS) branch, a high-frequency (HFQ) branch, and a CLR branch. Furthermore, we introduce a mutual learning (ML) mechanism to facilitate communication and learning among the branches, enabling them to take full advantage of all-round information in an effective and balanced manner. We evaluate the proposed method on three benchmark datasets, including CUHK-PEDES, ICFG-PEDES, and RSTPReid. The experimental results demonstrate that EAIBC significantly outperforms existing methods and achieves state-of-the-art (SOTA) performance in supervised, weakly supervised, and cross-domain settings.

图形着色的问题及其应用

图形着色的问题及其应用

Ising-Machine-Based Solver for Constrained Graph Coloring Problems

Ising-Machine-Based Solver for Constrained Graph Coloring Problems

Animation line art colorization based on the optical flow method

AbstractColoring an animation sketch sequence is a challenging task in computer vision since the information contained in line sketches is too sparse, and the colors need to be uniform between continuous frames. Many the existing colorization algorithms can only be applied to one image and can be considered color filling algorithms. Such algorithms only provide a color result that fits within a reasonable range and can not be applied to the coloring of frame sequences. This paper proposes an end‐to‐end two‐stage optical flow colorization network to solve the animation frame sequence colorization problem. The first stage of the network finds the direction of the color pixel flow from the detail change between a given reference frame and the next frame of line artwork and then completes the initial coloring process. The second stage of the network performs color correction and clarifies the output of the first stage. Since our algorithm does not directly colorize the image but finds the path of the color change to colorize it, it ensures a consistent color space for the sequence frames after colorization. We conduct experiments on an animation dataset, and the results show that our algorithm is effective. The code is available at https://github.com/silenye/Colorization.

Improving Children's Fine Motor and Aesthetic Abilities Through Coloring Activities with Various Media

Based on observations of development activities in Group B of the Aisyiyah Bustanul Athfal Sindang Indramayu Kindergarten, it was found that there was a problem of low children's coloring and aesthetic abilities. Where only around 4 children could complete it well or only around 26% of all children in group B. This research aims to improve children's fine motor and aesthetic physical abilities through coloring pictures using various media. This research uses a quantitative and qualitative descriptive design, with the Classroom Action Research (PTK) method which consists of several research steps, namely planning, implementation, observation and reflection. This research was carried out in two cycles, each cycle creating 5 RPPH and improvement plans. The results of the research show that children's fine motor and aesthetic abilities through coloring pictures using various media can be utilized in learning activities to develop children's fine motor and aesthetic physical abilities. In cycle 1, children's physical motor and aesthetic abilities increased from 26% to 46%. Then in cycle 2 it increased to 93%. This is characterized by an increase in children's ability to color pictures neatly, with good color combinations, in completing activities independently and so on, which also develops all aspects of children's development such as physical, motoric, cognitive, artistic, social emotional and behavioral.

Eventually lattice-linear algorithms

Lattice-linear systems allow nodes to execute asynchronously. We introduce eventually lattice-linear algorithms, where lattices are induced only among the states in a subset of the state space. The algorithm guarantees that the system transitions to a state in one of the lattices. Then, the algorithm behaves lattice linearly while traversing to an optimal state through that lattice.We present a lattice-linear self-stabilizing algorithm for service demand based minimal dominating set (SDMDS) problem. Using this as an example, we elaborate the working of, and define, eventually lattice-linear algorithms. Then, we present eventually lattice-linear self-stabilizing algorithms for minimal vertex cover (MVC), maximal independent set (MIS), graph colouring (GC) and 2-dominating set problems (2DS).Algorithms for SDMDS, MVC and MIS converge in 1 round plus n moves (within 2n moves), GC in n+4m moves, and 2DS in 1 round plus 2n moves (within 3n moves). These results are an improvement over the existing literature. We also present experimental results to show performance gain demonstrating the benefit of lattice-linearity.

On a colorful problem by Dol'nikov concerning translates of convex bodies

In this note we study a conjecture by Jerónimo-Castro, Magazinov and Soberón which generalized a question posed by Dol'nikov. Let F1,F2,…,Fn be families of translates of a convex compact set K in the plane so that each two sets from distinct families intersect. We show that, for some j, ⋃i≠jFi can be pierced by at most 4 points. To do so, we use previous ideas from Gomez-Navarro and Roldán-Pensado together with an approximation result closely tied to the Banach-Mazur distance to the square.

Impact of Shellac and Gum Arabic in Shoe Polish Formulation based on Karanj Oil

The study was conducted from August, 2021 to March, 2022 at the Department of Agricultural Engineering, Birsa Agricultural University, Kanke, Ranchi, Jharkhand, India to develop shoe polish based on Karanj oil and the impact of shellac and gum arabic in its formulation. The standardized composition of shoe polish was obtained as 3 part oil and one-part bee wax containing other elements like colour pigment, shellac, gum arabic and turpentine oil. To overcome the problem of colour, sixteen various combinations of shellac and gum arabic were carried out for improvement of colour in each case of black and brown polish. The developed shoe polish was tested for texture, applicability, spreadability, and ease in taking out the material. Colour was recorded with a gloss meter. The highest gloss meter reading i.e. 10.62 GU was recorded in the case of a black shoe polish containing shellac and gum arabic each weighing 0.75 g whereas a sample containing 1.25 g shellac and gum arabic each showed the lowest gloss reading of 3.88 GU. In the case of brown polish, the highest gloss meter reading i.e. 5.24 GU was recorded in the case of a sample containing shellac and gum arabic weighing 1.25 g each, and the lowest was observed in the case of the sample containing shellac weighing 0.5g and gum arabic weighing 0.75 g. Based on gloss value and sensory evaluation, the developed sample was found very near to standard brand shoe polish (Cherry) except that it gave an oily appearance initially.

Revisit the Coloring Problem of Gallai Graphs

Revisit the Coloring Problem of Gallai Graphs

How Blue Is Read: Language and Sensation in Literature and Philosophy

Abstract: Philosophers and art critics have long argued that the language of color misses or even mars the ineffable sensation of color. But a literary perspective shows otherwise. Starting with examples of colors read but not seen, and then discussing how philosophers have addressed (and often muddled) the so-called problem of color, I propose thinking of color terms as techniques for stabilizing and directing color sensations. I then show how William H. Gass and Maggie Nelson develop a version of this idea in their respective books about blue, which are really books about the relationship between writing and quality.

Iterated Clique Reductions in Vertex Weighted Coloring for Large Sparse Graphs.

The Minimum Vertex Weighted Coloring (MinVWC) problem is an important generalization of the classic Minimum Vertex Coloring (MinVC) problem which is NP-hard. Given a simple undirected graph G=(V,E), the MinVC problem is to find a coloring s.t. any pair of adjacent vertices are assigned different colors and the number of colors used is minimized. The MinVWC problem associates each vertex with a positive weight and defines the weight of a color to be the weight of its heaviest vertices, then the goal is the find a coloring that minimizes the sum of weights over all colors. Among various approaches, reduction is an effective one. It tries to obtain a subgraph whose optimal solutions can conveniently be extended into optimal ones for the whole graph, without costly branching. In this paper, we propose a reduction algorithm based on maximal clique enumeration. More specifically our algorithm utilizes a certain proportion of maximal cliques and obtains lower bounds in order to perform reductions. It alternates between clique sampling and graph reductions and consists of three successive procedures: promising clique reductions, better bound reductions and post reductions. Experimental results show that our algorithm returns considerably smaller subgraphs for numerous large benchmark graphs, compared to the most recent method named RedLS. Also, we evaluate individual impacts and some practical properties of our algorithm. Furthermore, we have a theorem which indicates that the reduction effects of our algorithm are equivalent to that of a counterpart which enumerates all maximal cliques in the whole graph if the run time is sufficiently long.

B-Coloring Parameterized by Clique-Width

We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial time results on graph classes, and answers open questions posed by Campos and Silva (Algorithmica 80(1), 104–115, 2018) and Bonomo et al. (Graphs and Combinatorics 25(2), 153–167, 2009). This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is FPT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{FPT}$$\\end{document} when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound. The running times of the clique-width based algorithms for b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b$$\\end{document}-Coloring and Fall Coloring are tight under the Exponential Time Hypothesis.

On structural parameterizations of load coloring

Given a graph G and a positive integer k, the 2-Load Coloring problem is to check whether there is a 2-coloring f:V(G)→{r,b} of G such that for every i∈{r,b}, there are at least k edges with both end vertices colored i. It is known that the problem is NP-complete even on special classes of graphs like regular graphs. Gutin and Jones (2014) showed that the problem is fixed-parameter tractable by giving a kernel with at most 7k vertices. Barbero et al. (2017) obtained a kernel with less than 4k vertices and O(k) edges, improving the earlier result.In this paper, we study the parameterized complexity of the problem with respect to structural graph parameters. We show that 2-Load Coloring cannot be solved in time f(w)no(w), unless ETH fails and it can be solved in time nO(w), where n is the size of the input graph, w is the clique-width of the graph and f is an arbitrary function of w. Next, we consider the parameters distance to cluster graphs, distance to co-cluster graphs and distance to threshold graphs, which are weaker than the parameter clique-width and show that the problem is fixed-parameter tractable (FPT) with respect to these parameters. Finally, we show that 2-Load Coloring is NP-complete even on bipartite graphs and split graphs.

Attention-based dual-color space fusion network for low-light image enhancement

The low visibility and dull colors associated with low-light images are not only difficult to satisfy the photographer, but also hinder further visual tasks. In this study, we propose an attention-based dual-color space fusion network to enhance low-light images. By introducing HSV color space, the network can solve the problems of dim color and insufficient contrast existing in previous RGB color space methods. In both color spaces, the network adopts a large number of attention mechanisms to highlight important features in real time. To better achieve feature fusion across color space, we design adaptive large-kernel attention (ALKA) and feature extraction module (FEM), respectively. The ALKA adaptively selects features for enriching the encoding input in the HSV color space during the RGB color space encoding process. The FEM assists the network in more comprehensive supervision of the enhancement process and generates the attention map to transmit the marking features in the previous color space. At the end of the network, we also design channel enhancement module (CEM) to protect the texture details lost during the scale change. We have conducted experiments on a variety of public datasets, and both subjective and objective comparisons have demonstrated the particularity and prominence of our method.