Graph Colorings
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The Vertex Coloring Problem and Its Generalizations
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by AMS Tesi di Dottorato UNIVERSITA` DEGLI STUDI DI BOLOGNA Dottorato di Ricerca in Automatica e Ricerca Operativa MAT/09 XIX Ciclo The Vertex Coloring Problem and its Generalizations Enrico Malaguti Il Coordinatore Il Tutor Prof. Claudio Melchiorri Prof. Paolo Toth A.A. 2003{2006 Contents Acknowledgments v Keywords vii List of ¯gures ix List of tables xi 1 Introduction 1 1.1 The Vertex Coloring Problem and its Generalizations . 1 1.2 Fair Routing . 6 I Vertex Coloring Problems 9 2 A Metaheuristic Approach for the Vertex Coloring Problem 11 2.1 Introduction . 11 2.1.1 The Heuristic Algorithm MMT . 12 2.1.2 Initialization Step . 13 2.2 PHASE 1: Evolutionary Algorithm . 15 2.2.1 Tabu Search Algorithm . 15 2.2.2 Evolutionary Diversi¯cation . 18 2.2.3 Evolutionary Algorithm as part of the Overall Algorithm . 21 2.3 PHASE 2: Set Covering Formulation . 22 2.3.1 The General Structure of the Algorithm . 24 2.4 Computational Analysis . 25 2.4.1 Performance of the Tabu Search Algorithm . 25 2.4.2 Performance of the Evolutionary Algorithm . 26 2.4.3 Performance of the Overall Algorithm . 26 2.4.4 Comparison with the most e®ective heuristic algorithms . 30 2.5 Conclusions . 33 3 An Evolutionary Approach for Bandwidth Multicoloring Problems 37 3.1 Introduction . 37 3.2 An ILP Model for the Bandwidth Coloring Problem . 39 3.3 Constructive Heuristics . 40 3.4 Tabu Search Algorithm . 40 i ii CONTENTS 3.5 The Evolutionary Algorithm . -
Indian Journal of Science ANALYSIS International Journal for Science ISSN 2319 – 7730 EISSN 2319 – 7749
Indian Journal of Science ANALYSIS International Journal for Science ISSN 2319 – 7730 EISSN 2319 – 7749 Graph colouring problem applied in genetic algorithm Malathi R Assistant Professor, Dept of Mathematics, Scsvmv University, Enathur, Kanchipuram, Tamil Nadu 631561, India, Email Id: [email protected] Publication History Received: 06 January 2015 Accepted: 05 February 2015 Published: 18 February 2015 Citation Malathi R. Graph colouring problem applied in genetic algorithm. Indian Journal of Science, 2015, 13(38), 37-41 ABSTRACT In this paper we present a hybrid technique that applies a genetic algorithm followed by wisdom of artificial crowds that approach to solve the graph-coloring problem. The genetic algorithm described here, utilizes more than one parent selection and mutation methods depending u p on the state of fitness of its best solution. This results in shifting the solution to the global optimum, more quickly than using a single parent selection or mutation method. The algorithm is tested against the standard DIMACS benchmark tests while, limiting the number of usable colors to the chromatic numbers. The proposed algorithm succeeded to solve the sample data set and even outperformed a recent approach in terms of the minimum number of colors needed to color some of the graphs. The Graph Coloring Problem (GCP) is also known as complete problem. Graph coloring also includes vertex coloring and edge coloring. However, mostly the term graph coloring refers to vertex coloring rather than edge coloring. Given a number of vertices, which form a connected graph, the objective is that to color each vertex so that if two vertices are connected in the graph (i.e. -
Restricted Colorings of Graphs
Restricted colorings of graphs Noga Alon ∗ Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel and Bellcore, Morristown, NJ 07960, USA Abstract The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of attention. Here we describe the techniques applied in the study of this subject, which combine combinatorial, algebraic and probabilistic methods, and discuss several intriguing conjectures and open problems. This is mainly a survey of recent and less recent results in the area, but it contains several new results as well. ∗Research supported in part by a United States Israel BSF Grant. Appeared in "Surveys in Combinatorics", Proc. 14th British Combinatorial Conference, London Mathematical Society Lecture Notes Series 187, edited by K. Walker, Cambridge University Press, 1993, 1-33. 0 1 Introduction Graph coloring is arguably the most popular subject in graph theory. An interesting variant of the classical problem of coloring properly the vertices of a graph with the minimum possible number of colors arises when one imposes some restrictions on the colors available for every vertex. This variant received a considerable amount of attention that led to several fascinating conjectures and results, and its study combines interesting combinatorial techniques with powerful algebraic and probabilistic ideas. The subject, initiated independently by Vizing [51] and by Erd}os,Rubin and Taylor [24], is usually known as the study of the choosability properties of a graph. In the present paper we survey some of the known recent and less recent results in this topic, focusing on the techniques involved and mentioning some of the related intriguing open problems. -
Parameterized Algorithms for Recognizing Monopolar and 2
Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs∗ Iyad Kanj1, Christian Komusiewicz2, Manuel Sorge3, and Erik Jan van Leeuwen4 1School of Computing, DePaul University, Chicago, USA, [email protected] 2Institut für Informatik, Friedrich-Schiller-Universität Jena, Germany, [email protected] 3Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany, [email protected] 4Max-Planck-Institut für Informatik, Saarland Informatics Campus, Saarbrücken, Germany, [email protected] October 16, 2018 Abstract A graph G is a (ΠA, ΠB)-graph if V (G) can be bipartitioned into A and B such that G[A] satisfies property ΠA and G[B] satisfies property ΠB. The (ΠA, ΠB)-Recognition problem is to recognize whether a given graph is a (ΠA, ΠB )-graph. There are many (ΠA, ΠB)-Recognition problems, in- cluding the recognition problems for bipartite, split, and unipolar graphs. We present efficient algorithms for many cases of (ΠA, ΠB)-Recognition based on a technique which we dub inductive recognition. In particular, we give fixed-parameter algorithms for two NP-hard (ΠA, ΠB)-Recognition problems, Monopolar Recognition and 2-Subcoloring, parameter- ized by the number of maximal cliques in G[A]. We complement our algo- rithmic results with several hardness results for (ΠA, ΠB )-Recognition. arXiv:1702.04322v2 [cs.CC] 4 Jan 2018 1 Introduction A (ΠA, ΠB)-graph, for graph properties ΠA, ΠB , is a graph G = (V, E) for which V admits a partition into two sets A, B such that G[A] satisfies ΠA and G[B] satisfies ΠB. There is an abundance of (ΠA, ΠB )-graph classes, and important ones include bipartite graphs (which admit a partition into two independent ∗A preliminary version of this paper appeared in SWAT 2016, volume 53 of LIPIcs, pages 14:1–14:14. -
A Survey of Graph Coloring - Its Types, Methods and Applications
FOUNDATIONS OF COMPUTING AND DECISION SCIENCES Vol. 37 (2012) No. 3 DOI: 10.2478/v10209-011-0012-y A SURVEY OF GRAPH COLORING - ITS TYPES, METHODS AND APPLICATIONS Piotr FORMANOWICZ1;2, Krzysztof TANA1 Abstract. Graph coloring is one of the best known, popular and extensively researched subject in the eld of graph theory, having many applications and con- jectures, which are still open and studied by various mathematicians and computer scientists along the world. In this paper we present a survey of graph coloring as an important subeld of graph theory, describing various methods of the coloring, and a list of problems and conjectures associated with them. Lastly, we turn our attention to cubic graphs, a class of graphs, which has been found to be very interesting to study and color. A brief review of graph coloring methods (in Polish) was given by Kubale in [32] and a more detailed one in a book by the same author. We extend this review and explore the eld of graph coloring further, describing various results obtained by other authors and show some interesting applications of this eld of graph theory. Keywords: graph coloring, vertex coloring, edge coloring, complexity, algorithms 1 Introduction Graph coloring is one of the most important, well-known and studied subelds of graph theory. An evidence of this can be found in various papers and books, in which the coloring is studied, and the problems and conjectures associated with this eld of research are being described and solved. Good examples of such works are [27] and [28]. In the following sections of this paper, we describe a brief history of graph coloring and give a tour through types of coloring, problems and conjectures associated with them, and applications. -
Star Coloring Outerplanar Bipartite Graphs
Discussiones Mathematicae Graph Theory 39 (2019) 899–908 doi:10.7151/dmgt.2109 STAR COLORING OUTERPLANAR BIPARTITE GRAPHS Radhika Ramamurthi and Gina Sanders Department of Mathematics California State University San Marcos San Marcos, CA 92096-0001 USA e-mail: [email protected] Abstract A proper coloring of the vertices of a graph is called a star coloring if at least three colors are used on every 4-vertex path. We show that all outerplanar bipartite graphs can be star colored using only five colors and construct the smallest known example that requires five colors. Keywords: chromatic number, star coloring, outerplanar bipartite graph. 2010 Mathematics Subject Classification: 05C15. 1. Introduction A proper r-coloring of a graph G is an assignment of labels from {1, 2,...,r} to the vertices of G so that adjacent vertices receive distinct colors. The minimum r so that G has a proper r-coloring is called the chromatic number of G, denoted by χ(G). The chromatic number is one of the most studied parameters in graph theory, and by convention, the term coloring of a graph is usually used instead of proper coloring. In 1973, Gr¨unbaum [5] considered proper colorings with the additional constraint that the subgraph induced by every pair of color classes is acyclic, i.e., contains no cycles. He called such colorings acyclic colorings, and the minimum r such that G has an acyclic r-coloring is called the acyclic chromatic number of G, denoted by a(G). In introducing the notion of an acyclic coloring, Gr¨unbaum noted that the condition that the union of any two color classes induces a forest can be generalized to other bipartite graphs. -
COLORING and DEGENERACY for DETERMINING VERY LARGE and SPARSE DERIVATIVE MATRICES ASHRAFUL HUQ SUNY Bachelor of Science, Chittag
COLORING AND DEGENERACY FOR DETERMINING VERY LARGE AND SPARSE DERIVATIVE MATRICES ASHRAFUL HUQ SUNY Bachelor of Science, Chittagong University of Engineering & Technology, 2013 A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge in Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE Department of Mathematics and Computer Science University of Lethbridge LETHBRIDGE, ALBERTA, CANADA c Ashraful Huq Suny, 2016 COLORING AND DEGENERACY FOR DETERMINING VERY LARGE AND SPARSE DERIVATIVE MATRICES ASHRAFUL HUQ SUNY Date of Defence: December 19, 2016 Dr. Shahadat Hossain Supervisor Professor Ph.D. Dr. Daya Gaur Committee Member Professor Ph.D. Dr. Robert Benkoczi Committee Member Associate Professor Ph.D. Dr. Howard Cheng Chair, Thesis Examination Com- Associate Professor Ph.D. mittee Dedication To My Parents. iii Abstract Estimation of large sparse Jacobian matrix is a prerequisite for many scientific and engi- neering problems. It is known that determining the nonzero entries of a sparse matrix can be modeled as a graph coloring problem. To find out the optimal partitioning, we have proposed a new algorithm that combines existing exact and heuristic algorithms. We have introduced degeneracy and maximum k-core for sparse matrices to solve the problem in stages. Our combined approach produce better results in terms of partitioning than DSJM and for some test instances, we report optimal partitioning for the first time. iv Acknowledgments I would like to express my deep acknowledgment and profound sense of gratitude to my supervisor Dr. Shahadat Hossain, for his continuous guidance, support, cooperation, and persistent encouragement throughout the journey of my MSc program. -
Sub-Coloring and Hypo-Coloring Interval Graphs⋆
Sub-coloring and Hypo-coloring Interval Graphs? Rajiv Gandhi1, Bradford Greening, Jr.1, Sriram Pemmaraju2, and Rajiv Raman3 1 Department of Computer Science, Rutgers University-Camden, Camden, NJ 08102. E-mail: [email protected]. 2 Department of Computer Science, University of Iowa, Iowa City, Iowa 52242. E-mail: [email protected]. 3 Max-Planck Institute for Informatik, Saarbr¨ucken, Germany. E-mail: [email protected]. Abstract. In this paper, we study the sub-coloring and hypo-coloring problems on interval graphs. These problems have applications in job scheduling and distributed computing and can be used as “subroutines” for other combinatorial optimization problems. In the sub-coloring problem, given a graph G, we want to partition the vertices of G into minimum number of sub-color classes, where each sub-color class induces a union of disjoint cliques in G. In the hypo-coloring problem, given a graph G, and integral weights on vertices, we want to find a partition of the vertices of G into sub-color classes such that the sum of the weights of the heaviest cliques in each sub-color class is minimized. We present a “forbidden subgraph” characterization of graphs with sub-chromatic number k and use this to derive a a 3-approximation algorithm for sub-coloring interval graphs. For the hypo-coloring problem on interval graphs, we first show that it is NP-complete and then via reduction to the max-coloring problem, show how to obtain an O(log n)-approximation algorithm for it. 1 Introduction Given a graph G = (V, E), a k-sub-coloring of G is a partition of V into sub-color classes V1,V2,...,Vk; a subset Vi ⊆ V is called a sub-color class if it induces a union of disjoint cliques in G. -
List of NP-Complete Problems from Wikipedia, the Free Encyclopedia
List of NP -complete problems - Wikipedia, the free encyclopedia Page 1 of 17 List of NP-complete problems From Wikipedia, the free encyclopedia Here are some of the more commonly known problems that are NP -complete when expressed as decision problems. This list is in no way comprehensive (there are more than 3000 known NP-complete problems). Most of the problems in this list are taken from Garey and Johnson's seminal book Computers and Intractability: A Guide to the Theory of NP-Completeness , and are here presented in the same order and organization. Contents ■ 1 Graph theory ■ 1.1 Covering and partitioning ■ 1.2 Subgraphs and supergraphs ■ 1.3 Vertex ordering ■ 1.4 Iso- and other morphisms ■ 1.5 Miscellaneous ■ 2 Network design ■ 2.1 Spanning trees ■ 2.2 Cuts and connectivity ■ 2.3 Routing problems ■ 2.4 Flow problems ■ 2.5 Miscellaneous ■ 2.6 Graph Drawing ■ 3 Sets and partitions ■ 3.1 Covering, hitting, and splitting ■ 3.2 Weighted set problems ■ 3.3 Set partitions ■ 4 Storage and retrieval ■ 4.1 Data storage ■ 4.2 Compression and representation ■ 4.3 Database problems ■ 5 Sequencing and scheduling ■ 5.1 Sequencing on one processor ■ 5.2 Multiprocessor scheduling ■ 5.3 Shop scheduling ■ 5.4 Miscellaneous ■ 6 Mathematical programming ■ 7 Algebra and number theory ■ 7.1 Divisibility problems ■ 7.2 Solvability of equations ■ 7.3 Miscellaneous ■ 8 Games and puzzles ■ 9 Logic ■ 9.1 Propositional logic ■ 9.2 Miscellaneous ■ 10 Automata and language theory ■ 10.1 Automata theory http://en.wikipedia.org/wiki/List_of_NP-complete_problems 12/1/2011 List of NP -complete problems - Wikipedia, the free encyclopedia Page 2 of 17 ■ 10.2 Formal languages ■ 11 Computational geometry ■ 12 Program optimization ■ 12.1 Code generation ■ 12.2 Programs and schemes ■ 13 Miscellaneous ■ 14 See also ■ 15 Notes ■ 16 References Graph theory Covering and partitioning ■ Vertex cover [1][2] ■ Dominating set, a.k.a. -
Combinatorial Optimization and Recognition of Graph Classes with Applications to Related Models
Combinatorial Optimization and Recognition of Graph Classes with Applications to Related Models Von der Fakult¨at fur¨ Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Mathematiker George B. Mertzios aus Thessaloniki, Griechenland Berichter: Privat Dozent Dr. Walter Unger (Betreuer) Professor Dr. Berthold V¨ocking (Zweitbetreuer) Professor Dr. Dieter Rautenbach Tag der mundlichen¨ Prufung:¨ Montag, den 30. November 2009 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.¨ Abstract This thesis mainly deals with the structure of some classes of perfect graphs that have been widely investigated, due to both their interesting structure and their numerous applications. By exploiting the structure of these graph classes, we provide solutions to some open problems on them (in both the affirmative and negative), along with some new representation models that enable the design of new efficient algorithms. In particular, we first investigate the classes of interval and proper interval graphs, and especially, path problems on them. These classes of graphs have been extensively studied and they find many applications in several fields and disciplines such as genetics, molecular biology, scheduling, VLSI design, archaeology, and psychology, among others. Although the Hamiltonian path problem is well known to be linearly solvable on interval graphs, the complexity status of the longest path problem, which is the most natural optimization version of the Hamiltonian path problem, was an open question. We present the first polynomial algorithm for this problem with running time O(n4). Furthermore, we introduce a matrix representation for both interval and proper interval graphs, called the Normal Interval Representation (NIR) and the Stair Normal Interval Representation (SNIR) matrix, respectively. -
The Harmonious Coloring Problem Is NP-Complete for Interval and Permutation Graphsଁ Katerina Asdre, Kyriaki Ioannidou, Stavros D
Discrete Applied Mathematics 155 (2007) 2377–2382 www.elsevier.com/locate/dam Note The harmonious coloring problem is NP-complete for interval and permutation graphsଁ Katerina Asdre, Kyriaki Ioannidou, Stavros D. Nikolopoulos Department of Computer Science, University of Ioannina, P.O. Box 1186, GR-45110 Ioannina, Greece Received 4 November 2006; received in revised form 25 May 2007; accepted 6 July 2007 Available online 14 September 2007 Abstract In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that the problem is also NP-complete for connected interval and permutation graphs. © 2007 Elsevier B.V. All rights reserved. Keywords: Harmonious coloring; Harmonious chromatic number; Achromatic number; Interval graphs; Permutation graphs; NP-completeness 1. Introduction Many NP-complete problems on arbitrary graphs admit polynomial time algorithms when restricted to the classes of interval graphs and cographs; NP-complete problems for these two classes of graphs that become solvable in polynomial time can be found in [1,3,7,12,15,16]. However, the pair-complete coloring problem, which is NP-hard on arbitrary graphs [17], remains NP-complete when restricted to graphs that are simultaneously interval and cographs [4].Apair- complete coloring of a simple graph G is a proper vertex coloring such that each pair of colors appears together on at least one edge, while the achromatic number (G) is the largest integer k for which G admits a pair-complete coloring with k colors. -
3-Coloring Gnp3 in Polynomial Expected Time
Mathematisch-Naturwissenschaftliche Fakultat¨ II Institut fur¨ Informatik Lehrstuhl fur¨ Algorithmen und Komplexit¨at Coloring sparse random k-colorable graphs in polynomial expected time Diplomarbeit eingereicht von: Julia B¨ottcher Gutachter: Dr. Amin Coja-Oghlan Dr. Mihyun Kang Berlin, den 5. Januar 2005 Erkl¨arung Hiermit erkl¨are ich, die vorliegende Arbeit selbstst¨andig verfasst und keine anderen als die angegebenen Hilfsmittel verwendet zu haben. Ich bin damit einverstanden, dass die vorliegen- de Arbeit in der Zentralbibliothek Naturwissenschaften der Humboldt-Universit¨at zu Berlin ¨offentlich ausgelegt wird. Berlin, den 5. Januar 2005 Julia B¨ottcher Zusammenfassung Nach wie vor stellt das Graphenf¨arbungsproblem eine der anspruchsvollsten algorithmischen Aufgaben innerhalb der Graphentheorie dar. So blieb bisher nicht nur die Suche nach effizienten Methoden, die dieses Problem exakt l¨osen, erfolglos. Gem¨aß eines Resultates von Feige und Kilian [22] ist auch die Existenz von signifikant besseren Approximationsalgorithmen als dem Trivialen, der jedem Knoten eine eigene Farbe zuordnet, sehr unwahrscheinlich. Diese Tatsache motiviert die Suche nach Algorithmen, die das Problem entweder nicht in allen sondern nur den meisten F¨allen entscheiden, dafur¨ aber von polynomieller Laufzeitkomplexit¨at sind, oder aber stets eine korrekte L¨osung ausgeben, eine polynomielle Laufzeit jedoch nur im Durchschnitt garantieren. Ein Algorithmus der ersterem Paradigma folgt, wurde von Alon und Kahale [2] fur¨ die fol- gende Klasse zuf¨alliger k-f¨arbbarer Graphen entwickelt: Konstruiere einen Graphen mit Gn,p,k einer Knotenmenge V der Kardinalit¨at n durch Partitionierung von V in k Mengen gleicher Gr¨oße und anschließendes Hinzufugen¨ der m¨oglichen Kanten zwischen diesen Mengen mit Wahrscheinlichkeit jeweils p.