Graph Coloring Research Articles

Fractal networks: Topology, dimension, and complexity.

Over the past two decades, the study of self-similarity and fractality in discrete structures, particularly complex networks, has gained momentum. This surge of interest is fueled by the theoretical developments within the theory of complex networks and the practical demands of real-world applications. Nonetheless, translating the principles of fractal geometry from the domain of general topology, dealing with continuous or infinite objects, to finite structures in a mathematically rigorous way poses a formidable challenge. In this paper, we overview such a theory that allows to identify and analyze fractal networks through the innate methodologies of graph theory and combinatorics. It establishes the direct graph-theoretical analogs of topological (Lebesgue) and fractal (Hausdorff) dimensions in a way that naturally links them to combinatorial parameters that have been studied within the realm of graph theory for decades. This allows to demonstrate that the self-similarity in networks is defined by the patterns of intersection among densely connected network communities. Moreover, the theory bridges discrete and continuous definitions by demonstrating how the combinatorial characterization of Lebesgue dimension via graph representation by its subsets (subgraphs/communities) extends to general topological spaces. Using this framework, we rigorously define fractal networks and connect their properties with established combinatorial concepts, such as graph colorings and descriptive complexity. The theoretical framework surveyed here sets a foundation for applications to real-life networks and future studies of fractal characteristics of complex networks using combinatorial methods and algorithms.

Some Results on the Rainbow Vertex-Disconnection Colorings of Graphs

Some Results on the Rainbow Vertex-Disconnection Colorings of Graphs

A Memetic Algorithm Designed for Solving Graph Coloring Problems: A Round-Robin Sports Scheduling Case Study

The issue of graph coloring is concerned with assigning colors to each vertex of an undirected graph so that adjacent vertices are not assigned the same color. Due to its NP-hard nature, a variety of heuristics and metaheuristics have been developed to tackle this problem. One of these approaches is the memetic algorithm, which was introduced as a solution for this problem. Another example is using a metaheuristic approach to address the round-robin sports scheduling problem. The algorithm suggested for this task comprises three primary components: (1) a randomized danger heuristic algorithm, (2) tabu search, and (3) a genetic algorithm with adaptive multi-parent crossover. To assess the performance of this algorithm, a set of test problems from different benchmark graphs in two categories were selected and compared with nine effective heuristics from existing literature. The results show that the algorithm performs exceptionally well on benchmark graphs that are known to be challenging. Additionally, a case study was conducted to demonstrate the effectiveness of the proposed algorithm

Equitable colorings of ��-corona products of cubic graphs

A graph G is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest integer k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by x=( G). In this paper the problem of determining the value of equitable chromatic number for multicoronas of cubic graphs G◦ lH is studied. The problem of ordinary coloring of multicoronas of cubic graphs is solvable in polynomial time. The complexity of equitable coloring problem is an open question for these graphs. We provide some polynomially solvable cases of cubical multicoronas and give simple linear time algorithms for equitable coloring of such graphs which use at most x=( G◦ lH) + 1 colors in the remaining cases.

Neighborhood Balanced Colorings of Graphs

Given a simple graph G, we ask when V(G) may be partitioned into two sets such that every vertex has an equal number of neighbors from each set. We establish a number of results for common families of graphs and completely classify 4-regular circulants which posses this property.

A SAT + Computer Algebra System Verification of the Ramsey Problem R(3, 8) (Student Abstract)

The Ramsey problem R(3,8) asks for the smallest n such that every red/blue coloring of the complete graph on n vertices must contain either a blue triangle or a red 8-clique. We provide the first certifiable proof that R(3,8) = 28, automatically generated by a combination of Boolean satisfiability (SAT) solver and a computer algebra system (CAS). This SAT+CAS combination is significantly faster than a SAT-only approach. While the R(3,8) problem was first computationally solved by McKay and Min in 1992, it was not a verifiable proof. The SAT+CAS method that we use for our proof is very general and can be applied to a wide variety of combinatorial problems.

New Results on the Robust Coloring Problem

Many variations of the classical graph coloring model have been intensively studied due to their multiple applications; scheduling problems and aircraft assignments, for instance, motivate the robust coloring problem. This model gets to capture natural constraints of those optimization problems by combining the information provided by two colorings: a vertex coloring of a graph and the induced edge coloring on a subgraph of its complement; the goal is to minimize, among all proper colorings of the graph for a fixed number of colors, the number of edges in the subgraph with the endpoints of the same color. The study of the robust coloring model has been focused on the search for heuristics due to its NP-hard character when using at least three colors, but little progress has been made in other directions. We present a new approach on the problem obtaining the first collection of non-heuristic results for general graphs; among them, we prove that robust coloring is the model that better approaches the equitable partition of the vertex set, even when the graph does not admit a so-called equitable coloring. We also show the NP-completeness of its decision problem for the unsolved case of two colors, obtain bounds on the associated robust coloring parameter, and solve a conjecture on paths that illustrates the complexity of studying this coloring model.

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.

Vertically Stackable Ovonic Threshold Switch Oscillator Using Atomic Layer Deposited Ge0.6Se0.4 Film for High-Density Artificial Neural Networks.

Nanodevice oscillators (nano-oscillators) have received considerable attention to implement in neuromorphic computing as hardware because they can significantly improve the device integration density and energy efficiency compared to complementary metal oxide semiconductor circuit-based oscillators. This work demonstrates vertically stackable nano-oscillators using an ovonic threshold switch (OTS) for high-density neuromorphic hardware. A vertically stackable Ge0.6Se0.4 OTS-oscillator (VOTS-OSC) is fabricated with a vertical crossbar array structure by growing Ge0.6Se0.4 film conformally on a contact hole structure using atomic layer deposition. The VOTS-OSC can be vertically integrated onto peripheral circuits without causing thermal damage because the fabrication temperature is <400 °C. The fabricated device exhibits oscillation characteristics, which can serve as leaky integrate-and-fire neurons in spiking neural networks (SNNs) and coupled oscillators in oscillatory neural networks (ONNs). For practical applications, pattern recognition and vertex coloring are demonstrated with SNNs and ONNs, respectively, using semiempirical simulations. This structure increases the oscillator integration density significantly, enabling complex tasks with a large number of oscillators. Moreover, it can enhance the computational speed of neural networks due to its rapid switching speed.

Deletion-contraction for a unified Laplacian and applications

We define a graph Laplacian with vertex weights in addition to the more classical edge weights, which unifies the combinatorial Laplacian and the normalised Laplacian. Moreover, we give a combinatorial interpretation for the coefficients of the weighted Laplacian characteristic polynomial in terms of weighted spanning forests and use this to prove a deletion-contraction relation. We prove various interlacing theorems relating to deletion and contraction, as well as to rectangular tilings, drawing on the work of Brooks, Smith, Stone and Tutte on square tilings. Additionally, we show that the weighted Laplacian also satisfies a vertex analogue of deletion-contraction. We give applications of weighted Laplacian eigenvalues to sparse cuts, independent sets and graph colouring, and establish new cases of a problem of Stanley on distinguishing nonisomorphic trees.

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.&#x0D; 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.&#x0D; METHODS: In this paper, we propose a novel method of implementing a Genetic Algorithm (GA) to address the GCP.&#x0D; 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.&#x0D; 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.

Canonical decompositions and algorithmic recognition of spatial graphs

AbstractWe prove that there exists an algorithm for determining whether two piecewise-linear spatial graphs are isomorphic. In its most general form, our theorem applies to spatial graphs furnished with vertex colourings, edge colourings and/or edge orientations.We first show that spatial graphs admit canonical decompositions into blocks, that is, spatial graphs that are non-split and have no cut vertices, in a suitable topological sense. Then, we apply a result of Haken and Matveev in order to algorithmically distinguish these blocks.

A graph neural network with negative message passing and uniformity maximization for graph coloring

AbstractGraph neural networks have received increased attention over the past years due to their promising ability to handle graph-structured data, which can be found in many real-world problems such as recommender systems and drug synthesis. Most existing research focuses on using graph neural networks to solve homophilous problems, and not much attention has been paid to heterophily-type problems. In this paper, we propose a graph network model for graph coloring, which is a class of representative heterophilous problems. Different from the message passing in conventional graph networks, we introduce negative message passing into a physics-inspired graph neural network for more effective information exchange in handling graph coloring problems. Moreover, a new term in the objective function taking into account the information entropy of nodes is suggested to increase the uniformity of the color assignment of each node, giving the neural network more chance to choose suitable colors for each node. Therefore, it could avoid the final solution getting stuck into the local optimum. Experimental studies are carried out to compare the proposed graph model with five state-of-the-art algorithms on ten publicly available graph coloring problems and d-regular graphs with up to $$10^4$$ 10 4 nodes, demonstrating the effectiveness of the proposed graph neural network.

Equitable coloring of planar graphs with maximum degree at least eight

The Chen-Lih-Wu Conjecture states that each connected graph with maximum degree Δ≥3 that is not the complete graph KΔ+1 or the complete bipartite graph KΔ,Δ admits an equitable coloring with Δ colors. For planar graphs, the conjecture has been confirmed for Δ≥13 by Yap and Zhang and for 9≤Δ≤12 by Nakprasit. In this paper, we present a proof that confirms the conjecture for graphs embeddable into a surface with non-negative Euler characteristic with maximum degree Δ≥9 and for planar graphs with maximum degree Δ≥8.

Tight Bounds for the N₂-Chromatic Number of Graphs

Let $G$ be a connected graph. A vertex coloring of $G$ is an $N_2$-vertex coloring if, for every vertex $v$, the number of different colors assigned to the vertices adjacent to $v$ is at most two. The $N_2$-chromatic number of $G$ is the maximum number of colors that can be used in an $N_2$-vertex coloring of $G$. In this paper, we establish tight bounds for the $N_2$-chromatic number of a graph in terms of its maximum degree and its diameter, and characterize those graphs that attain these bounds.

First Fit Algorithm: A Graph Coloring Approach to Conflict-Free University Course Timetabling

Aims: Tackling scheduling issues with the most optimal graph coloring algorithm has consistently posed significant difficulties. The university scheduling problem can be expressed as a graph coloring problem, where courses are depicted as vertices and the connections between courses that have common students or teachers are represented as edges. Subsequently, the task at hand is to assign the vertices with the minimum number of colors. In order to accomplish this task, this paper present a graph coloring technique to conflict free university course timetabling using first fit algorithms.&#x0D; Methodology: The conflict graph is partitioned into a set of independent color classes to be assigned time slots and transformed into a conflict-free timetable. The Ladoke Akintola University of Technology (LAUTECH) University Course Timetabling Data was adopted. The allocation of venue based on the allotted time slots is done using first fit packing algorithm. The proposed model is implemented using Python programming language. The developed model had courses being represented as vertices and edges. The course conflict graph was created based on the acquired dataset using vertices-edges relationship diagram. The implemented model is evaluated in terms of Halstead complexity metrics: Program Volume (PV), Program Length (PL), Program Effort (PE), Program Difficulty (PD) and Execution Time (ET). The PV, PL, PE, PD and ET values obtained for the implemented model are 18.45kbits, 0.51, 1037684, 1.97 and 20.45 secs, respectively.&#x0D; Conclusion: The proposed model shows a significant improvement over the existing models by producing conflict-free course timetabling problem with better evaluation results. This work will be highly useful in solving various scheduling, optimization and NP-hard related computational problems.

Finding Optimal Solutions with Neighborly Help

AbstractCan we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighbor instances, that is, instances with one local modification? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems, most notably, graph theory’s core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted). We focus on two prototypical graph problems, colorability and vertex cover. For example, we show that it is $$\text {NP}$$ NP -hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in $$\text {P}$$ P . We observe that vertex cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level. Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal-3-UnColorability is complete for $$\text {DP}$$ DP (differences of $$\text {NP}$$ NP sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For vertex cover, we show that recognizing $$\beta $$ β -vertex-critical graphs is complete for $$\Theta _2^\text {p}$$ Θ 2 p (parallel access to $$\text {NP}$$ NP ), obtaining the first completeness result for a criticality problem for this class.

Approximation algorithms for job scheduling with block-type conflict graphs

The problem of scheduling jobs on parallel machines (identical, uniform, or unrelated), under incompatibility relation modeled as a block graph, under the makespan optimality criterion, is considered in this paper. No two jobs that are in the relation (equivalently in the same block) may be scheduled on the same machine in this model.The presented model stems from a well-established line of research combining scheduling theory with methods relevant to graph coloring. Recently, cluster graphs and their extensions like block graphs were given additional attention. We complement hardness results provided by other researchers for block graphs by providing approximation algorithms. In particular, we provide a 2-approximation algorithm for P|G=block graph|Cmax and a PTAS for the case when the jobs are unit time in addition. In the case of uniform machines, we analyze two cases. The first one is when the number of blocks is bounded, i.e. Q|G=k − block graph|Cmax. For this case, we provide a PTAS, improving upon results presented by D. Page and R. Solis-Oba. The improvement is two-fold: we allow richer graph structure, and we allow the number of machine speeds to be part of the input. Due to strong NP-hardness of Q|G=2 − clique graph|Cmax, the result establishes the approximation status of Q|G=k − block graph|Cmax. The PTAS might be of independent interest because the problem is tightly related to the NUMERICAL k-DIMENSIONAL MATCHING WITH TARGET SUMS problem. The second case that we analyze is when the number of blocks is arbitrary, but the number of cut-vertices is bounded and jobs are of unit time. In this case, we present an exact algorithm. In addition, we present an FPTAS for graphs with bounded treewidth and a bounded number of unrelated machines.The paper ends with extensive tests of the selected algorithms.

Nordhaus–Gaddum-Type Results for the Strong Equitable Vertex k-Arboricity of Graphs

For a graph [Formula: see text] and positive integers [Formula: see text], [Formula: see text], a [Formula: see text]-tree-vertex coloring of [Formula: see text] refers to a [Formula: see text]-vertex coloring of [Formula: see text] satisfying every component of each induced subgraph generated by every set of vertices with the same color forms a tree with maximum degree not larger than [Formula: see text], and it is called equitable if the difference between the cardinalities of every pair of sets of vertices with the same color is at most [Formula: see text]. The strong equitable vertex [Formula: see text]-arboricity of [Formula: see text], denoted by [Formula: see text], is defined as the least positive integer [Formula: see text] satisfying [Formula: see text], which admits an equitable [Formula: see text]-tree-vertex coloring for each integer [Formula: see text] with [Formula: see text]. The strong equitable vertex [Formula: see text]-arboricity of a graph is very useful in graph theory applications such as load balance in parallel memory systems, constructing timetables and scheduling. In this paper, we present the tight upper and lower bounds on [Formula: see text] for an arbitrary graph [Formula: see text] with [Formula: see text] vertices and a given integer [Formula: see text] with [Formula: see text], and we characterize the extremal graphs [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text], respectively. Based on the above extremal results, we further obtain the Nordhaus–Gaddum-type results for [Formula: see text] of graphs [Formula: see text] with [Formula: see text] vertices for a given integer [Formula: see text] with [Formula: see text].

Proper Colorings in R-Regular Randic Index Graphs

Abstract: The new idea of proper colourings in the r-regular Randic index graph has been proposed in this paper. New Chromatic number inequalities connected to the Randic index are established in this paper.