Graph Coloring Research Articles

Chromatic quasisymmetric class functions for combinatorial Hopf monoids

We study the chromatic quasisymmetric class function of a linearized combinatorial Hopf monoid. Given a linearized combinatorial Hopf monoid H, and an H-structure h on a set N, there are proper colorings of h, generalizing graph colorings and poset partitions. We show that the automorphism group of h acts on the set of proper colorings. The chromatic quasisymmetric class function enumerates the fixed points of this action, weighting each coloring with a monomial. For the Hopf monoid of graphs this invariant generalizes Stanley’s chromatic symmetric function and specializes to the orbital chromatic polynomial of Cameron and Kayibi. We deduce various inequalities for the associated orbital polynomial invariants. We apply these results to several examples related to enumerating graph colorings, poset partitions, generic functions on matroids or generalized permutohedra, and others.

Proper conflict-free coloring of sparse graphs

A proper conflict-freec-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing exactly once on its neighborhood. This notion was formally introduced by Fabrici et al., who proved that planar graphs have a proper conflict-free 8-coloring and constructed a planar graph with no proper conflict-free 5-coloring. Caro, Petruševski, and Škrekovski investigated this coloring concept further, and in particular studied upper bounds on the maximum average degree that guarantees a proper conflict-free c-coloring for c∈{4,5,6}.Along these lines, we completely determine the threshold on the maximum average degree of a graph G, denoted mad(G), that guarantees a proper conflict-free c-coloring for all c and also provide tightness examples. Namely, for c≥5 we prove that a graph G with mad(G)≤4cc+2 has a proper conflict-free c-coloring, unless G contains a 1-subdivision of the complete graph on c+1 vertices. When c=4, we show that a graph G with mad(G)<125 has a proper conflict-free 4-coloring, unless G contains an induced 5-cycle. In addition, we show that a planar graph with girth at least 5 has a proper conflict-free 7-coloring.

Review Article: Graph Colouring and Applications

Review Article: Graph Colouring and Applications

PEMBAGIAN TUGAS APEL PAGI MINGGUAN PEGAWAI INSTANSI XYZ DENGAN MENGGUNAKAN PEWARNAAN GRAF BESERTA ANALISIS BEBAN PENUGASAN

This study addresses the issue of the weekly morning roll call duty assignment at XYZ Agency. The allocation of morning roll call duties at XYZ Agency has been incidental, leading to uneven task assignments for employees. The author proposes a solution to this problem by implementing the Welch-Powell Algorithm on Graph Coloring to determine the allocation of morning roll call duties for XYZ Agency employees. This is achieved by representing each employee as a point, and the edges represent job similarities. The graph coloring results in this study produce 4 colors: red, green, blue, and purple, with connected points having different colors. The findings reveal that 20 employees participate in the roll call duty each month. The duty assignments are then made for six months and compiled into a task assignment table. Based on descriptive statistical results, each employee receives an average assignment of 4.68 times with a standard deviation of 1.38. Furthermore, the Levene's Test is employed to ensure the equality of workload among different employee job groups. The results indicate differences in the assignment workload among job groups

Application of Graph Theory and Variants of Greedy Graph Coloring Algorithms for Optimization of Distributed Peer-to-Peer Blockchain Networks

This paper investigates the application of graph theory and variants of greedy graph coloring algorithms for the optimization of distributed peer-to-peer networks, with a special focus on private blockchain networks. The graph coloring problem, as an NP-hard problem, presents a challenge in determining the minimum number of colors needed to efficiently allocate resources within the network. The paper deals with the influence of different graph density, i.e., the number of links, on the efficiency of greedy algorithms such as DSATUR, Descending, and Ascending. Experimental results show that increasing the number of links in the network contributes to a more uniform distribution of colors and increases the resistance of the network, whereby the DSATUR algorithm achieves the most uniform color saturation. The optimal configuration for a 100-node network has been identified at around 2000 to 2500 links, which achieves stability without excessive redundancy. These results are applied in the context of a private blockchain network that uses optimal connectivity to achieve high resilience and efficient resource allocation. The research findings suggest that adapting network configuration using greedy algorithms can contribute to the optimization of distributed systems, making them more stable and resilient to loads.

A note on adjacent vertex distinguishing edge coloring of graphs

A note on adjacent vertex distinguishing edge coloring of graphs

Johan Coloring of Neighbourhood Corona of Graphs

Johan coloring or J-Coloring is the proper vertex coloring in which every vertices of a graph should belongs to rainbow neighbourhood . In a given graph G, a vertex v is said to be rainbow neighbourhood if every color class of G consists of atleast of one vertex of G from the closed neighbourhood. The Neighbourhood Corona is obtained by taking one copy of G1 and n1 copies of G2 and by joining each neighbour ith copy of G1 to each and every vertex of ith copy of G2. In this paper we determine Johan Chromatic number of Neighbourhood Coronna of Graphs.

On the list injective coloring of planar graphs without a $ {4^ - } $-cycle intersecting with a $ {5^ - } $-cycle

&lt;p&gt;An injective coloring of a graph $ G $ is a vertex coloring such that a pair of vertices obtain distinct colors if there is a path of length two between them. It is proved in this paper that $ \chi _i^l(G) \le \Delta + 4 $ if $ \Delta \ge 12 $ when $ G $ does not have a $ {4^ - } $-cycle intersecting with a $ {5^ - } $-cycle. Our result improves a previous result of Cai et al. in 2023, who showed that $ \chi _i^l(G) \le \Delta + 4 $ when $ \Delta \ge 12 $ and $ G $ has disjoint $ {5^ - } $-cycles.&lt;/p&gt;

Hoffman colorings of graphs

Hoffman colorings of graphs

Independence, matching and packing coloring of the iterated Mycielskian of graphs

Let α(G), ν(G), ν2(G) and χρ(G) denote respectively the independence, matching, 2-matching and packing chromatic numbers of a graph G. A well-known construction on graphs, called the Mycielskian of a graph, transforms any k-chromatic graph G into a (k+1)-chromatic graph M(G) having an equal clique number to G. The tth iterated Mycielskian of a graph G, denoted Mt(G), is obtained by iteratively repeating the Mycielskian transformation t times. In this paper, we give α(M(G)) in terms of ν2(G). Then we show that for all t≥2, α(Mt(G))=max{|Mt−1(G)|,2t−1α(M(G))}. We characterize for all t≥1, the connected graphs having α(Mt(G))=|Mt−1(G)| and those having α(Mt(G))=2tα(G). Afterwards, we give ν(Mt(G)) and ν2(Mt(G)) for all t≥1 in terms of ν2(G). Then we show that for all t≥1, Mt(G) is a König-Egerváry graph if and only if G does not have a perfect 2-matching. Later, we investigate the packing chromatic number of Mt(G). We present several sharp upper and lower bounds for χρ(Mt(G)), some in terms of the number of iterations t, the order of G, the k-independence number with k∈{1,2,3} and χρ(G). We show that χρ(Mt(G)) can be computed in polynomial time if G has a diameter at most 2. Recently, in Bidine et al. (2023) the authors studied diameter two graphs having χρ(Mt(G))=2tχρ(G) for t≥1. Here we fully characterize the diameter two graphs for which this equality holds. They also asked a question about the growth of χρ(Mt(G)) in terms of χρ(G). We show that for t≥1, χρ(Mt(G)) cannot be upper bounded by a function of χρ(G) alone. In addition, we discuss the realizable values for χρ(Mt(G)) and characterize the graphs having the least possible χρ(Mt(G)).

Classification, Applications and Future Directions of Bio-Inspired Algorithms From Swarm Intelligence to Quantum Computing Integration Using PROMETHEE Methodology

Biologically inspired algorithms (BIAs) are computational methods that model natural processes and systems. Although various types of BIAs have been developed over time, their classification has not been extensively studied. The most widely recognized types include evolution-based algorithms that draw inspiration from natural evolution and swarm-based algorithms that mimic the collective behaviors of animal groups. In addition, ecosystem-based algorithms provide another classification perspective. Many modern BIAs go beyond this well-established framework. These algorithms have shown themselves to be highly adaptable, finding application in a variety of domains to solve challenging optimization and decision-making problems. Examples include solving problems such as network routing, resource planning, graph coloring, and the traveling salesman problem. In artificial intelligence, BIAs have improved neural networks by improving clustering, classification, and prediction accuracy. In the medical field, they have made significant contributions to advances in diagnosis and treatment planning. As BIAs continue to evolve, their integration with advanced technologies such as quantum computing paves the way to overcome challenges such as slow integration and computational inefficiency. Reflecting the adaptability and resilience of natural systems, BIAs are increasingly being used in emerging areas such as cloud computing and wireless sensor networks, highlighting their growing relevance for scalable and reliable problem solving. Research significance: The focus of this research is on investigating biologically inspired algorithms (BIAs), which are computational techniques that model natural processes. These algorithms are crucial for solving complex optimization and decision-making problems in diverse domains such as resource planning, networking, artificial intelligence, and medicine. Despite their widespread applications, the classification of BIAs has received little attention. This research highlights the importance of understanding their classification and categorization, which can lead to more effective algorithm selection and optimization. By reviewing and refining the evolutionary-based, swarm-based, and ecosystem-based algorithm categories, this work helps to improve the theoretical framework of BIAs. In addition, the research explores the potential for improving BIA convergence rates by integrating emerging technologies such as quantum computing, which addresses key challenges in real-world applications. In the end, this research could promote the application of bio-inspired algorithms across a range of domains, enhancing computer efficiency, scalability, and flexibility. Methology: Alternatives: Genetic Algorithm (Evolutionary based), Particle Swarm Optimization (Swarm-based), Ant Colony Optimization (Swarm-based), Artificial Bee Colony Algorithm (Swarm-based). Evaluation Parameters: Convergence Speed, Accuracy, Scalability, Resource Efficiency, Computational Complexity, Energy Consumption, Noise Sensitivity. Result: The results show that Genetic Algorithm (Evolutionary based) received the highest ranking, whereas Artificial Bee Colony Algorithm (Swarm-based) received the lowest ranking. Conclusion: According to the PROMETHEE method, Genetic Algorithm (Evolutionary based) ranks highest in terms of its value for Bio-Inspired Algorithms.

Further Results on Equitable Near Proper Coloring of Derived Graph Families

Further Results on Equitable Near Proper Coloring of Derived Graph Families

Vertex Coloring and Eulerian and Hamiltonian Paths of Delaunay Graphs Associated with Sensor Networks

In this paper, we explore the connection between sensor networks and graph theory. Sensor networks represent distributed systems of interconnected devices that collect and transmit data, while graph theory provides a robust framework for modeling and analyzing complex networks. Specifically, we focus on vertex coloring, Eulerian paths, and Hamiltonian paths within the Delaunay graph associated with a sensor network. These concepts have critical applications in sensor networks, including connectivity analysis, efficient data collection, route optimization, task scheduling, and resource management. We derive theoretical results related to the chromatic number and the existence of Eulerian and Hamiltonian trails in the graph linked to the sensor network. Additionally, we complement this theoretical study with the implementation of several algorithmic procedures. A case study involving the monitoring of a sugarcane field, coupled with a computational analysis, demonstrates the performance and practical applicability of these algorithms in real-world scenarios.

On Bridge Graphs with Local Antimagic Chromatic Number 3

Let G=(V,E) be a connected graph. A bijection f:E→{1,…,|E|} is called a local antimagic labeling if, for any two adjacent vertices x and y, f+(x)≠f+(y), where f+(x)=∑e∈E(x)f(e), and E(x) is the set of edges incident to x. Thus, a local antimagic labeling induces a proper vertex coloring of G, where the vertex x is assigned the color f+(x). The local antimagic chromatic number χla(G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present some families of bridge graphs with χla(G)=3 and give several ways to construct bridge graphs with χla(G)=3.

Perfect Coloring of Graphs Related to Irreducible Fullerenes in Carbon Structures

Fullerenes are polyhedral molecules composed solely of carbon atoms, available in various sizes and shapes. These structures can also be depicted as graphs, with the vertices symbolizing the atoms and the edges representing the bonds between them. A fullerene graph is defined as a 3-connected, 3-regular planar graph that consists only of pentagonal and hexagonal faces. This paper examines the perfect 2- and 3-coloring of fullerene graphs, with a particular focus on irreducible fullerenes. The proposed approach begins by obtaining the adjacency matrix of the graphs and then comparing its eigenvalues with those of the parameter matrices. If the eigenvalues of a parameter matrix are a subset of the graph's eigenvalues, we retain these matrices for further analysis to determine their suitability for perfect coloring.

Neighbor full sum distinguishing total coloring of planar graphs

Neighbor full sum distinguishing total coloring of planar graphs

On asymmetric colourings of graphs with bounded degrees and infinite motion

On asymmetric colourings of graphs with bounded degrees and infinite motion

Equitable List Coloring of Planar Graphs With Given Maximum Degree

ABSTRACTIf is a list assignment of colors to each vertex of an ‐vertex graph , then an equitable ‐coloring of is a proper coloring of vertices of from their lists such that no color is used more than times. A graph is equitably ‐choosable if it has an equitable ‐coloring for every ‐list assignment . In 2003, Kostochka, Pelsmajer, and West (KPW) conjectured that an analog of the famous Hajnal–Szemerédi Theorem on equitable coloring holds for equitable list coloring, namely, that for each positive integer every graph with maximum degree at most is equitably ‐choosable. The main result of this paper is that for each and each planar graph , a stronger statement holds: if the maximum degree of is at most , then is equitably ‐choosable. In fact, we prove the result for a broader class of graphs—the class of the graphs in which each bipartite subgraph with has at most edges. Together with some known results, this implies that the KPW Conjecture holds for all graphs in , in particular, for all planar graphs. We also introduce the new stronger notion of strongly equitable (SE, for short) list coloring and prove all bounds for this parameter. An advantage of this is that if a graph is SE ‐choosable, then it is both equitably ‐choosable and equitably ‐colorable, while neither of being equitably ‐choosable and equitably ‐colorable implies the other.

Techniques for learning sparse Pauli-Lindblad noise models

Error-mitigation techniques such as probabilistic error cancellation and zero-noise extrapolation benefit from accurate noise models. The sparse Pauli-Lindblad noise model is one of the most successful models for those applications. In existing implementations, the model decomposes into a series of simple Pauli channels with one- and two-local terms that follow the qubit topology. While the model has been shown to accurately capture the noise in contemporary superconducting quantum processors for error mitigation, it is important to consider higher-weight terms and effects beyond nearest-neighbor interactions. For such extended models to remain practical, however, we need to ensure that they can be learned efficiently. In this work we present new techniques that accomplish exactly this. We introduce twirling based on Pauli rotations, which enables us to automatically generate single-qubit learning correction sequences and reduce the number of unique fidelities that need to be learned. In addition, we propose a basis-selection strategy that leverages graph coloring and uniform covering arrays to minimize the number of learning bases. Taken together, these techniques ensure that the learning of the extended noise models remains efficient, despite their increased complexity.

Airport Flight-Gate Allocation Assignment Using Local Antimagic Vertex Coloring

The management of aviation gate assignments may significantly affect the overall productivity of an airport. This work presents a novel approach to enhancing gate assignment algorithms. The methodology used in this study incorporates graph theory and other intricate techniques, with specific focus on the concept of local antimagic vertex coloring. This study presents a comprehensive methodology for identifying optimal gate locations, which incorporates several factors such as aircraft characteristics, passenger capacity, and resource availability. Based on the results of this study, the incorporation of local antimagic vertex coloring exhibits the capacity to significantly enhance the operational effectiveness of airports via the reduction of passenger line wait times and alleviation of traffic congestion challenges. The aforementioned result was derived by using comprehensive models of the physical cosmos and conducting a meticulous analysis of empirical facts. This novel concept has the capacity to enhance the efficacy of airport gate allocation, resulting in heightened levels of customer contentment and reduced operational costs. The implementation of this procedure will facilitate the preservation of the efficiency of the operational networks for air transportation.