Face-Wise Chromatic Number Cat Myrant
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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. -
A Study of the Upper Domatic Number of a Graph
A STUDY OF THE UPPER DOMATIC NUMBER OF A GRAPH A Thesis by NICHOLAS PHILLIPS Submitted to the Graduate School at Appalachian State University in partial fulfillment of the requirements for the degree of MASTER OF COMPUTER SCIENCE May 2017 Department of Computer Science A STUDY OF THE UPPER DOMATIC NUMBER OF A GRAPH A Thesis by NICHOLAS PHILLIPS May 2017 APPROVED BY: Alice McRae, Ph.D Chairperson, Thesis Committee Raghuveer Mohan, Ph.D Member, Thesis Committee Rahman Tashakkori, Ph.D Member, Thesis Committee James Wilkes, Ph.D Chairperson, Department of Computer Science Max C. Poole, Ph.D Dean, Cratis D. Williams School of Graduate Studies Copyright by Nicholas Phillips c 2017 All Rights Reserved Abstract A STUDY OF THE UPPER DOMATIC NUMBER OF A GRAPH Nicholas Phillips B.S., Appalachian State University M.S., Appalachian State University Chairperson: Alice McRae Given a graph G we can partition the vertices of G into k disjoint sets represented as π = fV1;V2;:::;Vkg: We say a set A of vertices dominates another set of vertices, B, if for every vertex b 2 B there exists some vertex a 2 A adjacent to b. The upper domatic number of a graph G is written D(G) and defined as the maximum integer k such that G can be partitioned into k sets where for every pair of sets Vi;Vj 2 π either Vi dominates Vj or Vj dominates Vi or both. In this thesis we introduce the upper domatic number of a graph and provide various results on the properties of the upper domatic number, notably that D(G) ≤ ∆(G), as well as relating it to other well-studied graph properties such as the achromatic, pseudoachromatic, and transitive numbers. -
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. -
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. -
REGULAR TOTALLY DOMATICALLY FULL GRAPHS Bohdan ZELINKA
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 86 (1990) 71-79 71 North-Holland REGULAR TOTALLY DOMATICALLY FULL GRAPHS Bohdan ZELINKA Katedra Mathematiky V&ST, Sokolska’ 8, 46001 Liberec 1, Czechoslovakia Received 2 December 1988 The paper studies bipartite undirected graphs and directed graphs which are regular and totally domatically full. 1. Introduction The domatic number of a graph was defined by Cockayne and Hedetniemi [4] and the total domatic number by the same authors and Dawes [3]; the total domatic number was studied in [l]. These concepts were defined for undirected graphs. The concept of the domatic number was transferred to directed graphs in [5]. Here we shall transfer also the concept of the total domatic number to directed graphs; for this goal we shall first study bipartite undirected graphs. All considered graphs are finite without loops and multiple edges. Fundamental results concerning the domination in graphs can be found in Berge’s book [2]. A dominating (or totally dominating) set in an undirected graph G is a subset D of the vertex set V(G) of G with the property that for each vertex x E V(G) - D (or x E V(G) respectively) there exists a vertex y E D adjacent to X. A domatic (or total domatic) partition of G is a partition of V(G), all of whose classes are dominating (or totally dominating respectively) sets in G. The domatic (or total domatic) number d(G) (or d,(G)) of G is the maximum number of classes of a domatic (or total domatic respectively) partition of G. -
Approximating the Independent Domatic Partition Problem in Random Geometric Graphs – an Experimental Study
CCCG 2010, Winnipeg MB, August 9–11, 2010 Approximating the Independent Domatic Partition Problem in Random Geometric Graphs – An Experimental Study Dhia Mahjoub ∗ Angelika Leskovskaya † David W. Matula ∗ Abstract partition a graph G into disjoint independent dominat- ing sets. The independent domatic number dind(G) is We investigate experimentally the Domatic Partition the maximum size of such a partition. (DP) problem, the Independent Domatic Partition For any graph G, dind(G) ≤ d(G) ≤ δ(G) + 1 where (IDP) problem and the Idomatic partition problem in δ(G) denotes the minimum degree in G. If d(G) = Random Geometric Graphs (RGGs). In particular, δ(G)+1 and/or dind(G) = δ(G)+1, then G is called do- we model these problems as Integer Linear Programs matically full (DF) and/or independent domatically full (ILPs), solve them optimally, and show on a large set (IDF) respectively [3]. A graph whose vertices V can of samples that RGGs are independent domatically full be strictly partitioned into disjoint independent domi- most likely (over 93% of the cases) and domatically full nating sets is termed indominable [1] or idomatic [3]. almost certainly (100% of the cases). We empirically The idomatic number id(G) is the partition’s maximum confirm using two methods that RGGs are not idomatic size. Notice that the DP, IDP and idomatic problems on any of the samples. We compare the results of the are all NP-complete in general graphs [2, 4, 8] and also ILP-based exact algorithms with those of known color- believed to be so in UDGs [13]. -
Domatically Critical and Domatically Full Graphs
Discrete Mathematics 86 (1990) 81-87 81 North-Holland DOMATICALLY CRITICAL AND DOMATICALLY FULL GRAPHS Douglas F. RALL Department of Mathematics, Furman University, Greenville, SC 29613, USA Received 2 December 1988 A subset, D, of the vertex set of a graph G is called a dominating set of G if each vertex of G is either in D or adjacent to some vertex in D. The maximum cardinality of a partition of the vertex set of G into dominating sets is the domatic number of G, denoted d(G). G is said to be domatically critical if the removal of any edge of G decreases the domatic number, and G is domatically full if d(G) assumes the known lower bound of 6(G) + 1. An example is given to settle a conjecture of B. Zelinka concerning the structure of a domatically critical graph. We also prove that a domatically critical graph G is domatically full if d(G) < 3 and provide’ examples to show this does not extend to the cases d(G) > 3. 1. Introduction Consider a finite graph G with vertex set V(G) and edge set E(G), which has neither loops nor multiple edges. For x E V(G), N(x) will denote the set of vertices in V(G) each of which is adjacent to X. If G and H are two graphs having no vertices in common, then G + H denotes the join of G and H. G + H has vertex set V(G) U V(H) and edge set E(G) U E(H) U {xy (x E V(G) and y E V(H)}. -
Complexity of the Exact Domatic Number Problem and of the Exact
Electronic Colloquium on Computational Complexity, Report No. 68 (2002) Complexity of the Exact Domatic Number Problem and of the Exact Conveyor Flow Shop Problem ¡ Tobias Riege and Jor¨ g Rothe Institut fur¨ Informatik Heinrich-Heine-Universitat¨ Dusseldorf¨ 40225 Dusseldorf,¨ Germany December 9, 2002 Abstract We prove that the exact versions of the domatic number problem are complete for the levels of the boolean hierarchy over NP. The domatic number problem, which arises in the area of computer networks, is the problem of partitioning a given graph into a maximum number of disjoint dominating sets. This number is called the domatic number of the graph. We prove that the problem of determining whether or not the domatic number of a given graph is £¥¤§¦ ¨ © ¢ exactly one of ¢ given values is complete for BH NP , the th level of the boolean hierarchy over NP. In particular, for ¢ , it is DP-complete to determine whether or not the domatic £¦ ¨ number of a given graph equals exactly a given integer. Note that DP BH NP . We obtain similar results for the exact versions of the conveyor flow shop problem, which arises in real- world applications in the wholesale business, where warehouses are supplied with goods from a central storehouse. Our reductions apply Wagner’s conditions sufficient to prove hardness for the levels of the boolean hierarchy over NP. Key words: Computational complexity; completeness; domatic number problem; conveyor flow shop problem; boolean hierarchy !"#!$%& ! Email: - . ' !"#!$%& ! Email: $() - . ISSN 1433-8092 1 Introduction ¡ ¢¤£¥ §¦ A dominating set in an undirected graph is a subset of the vertex set such that ¡ ¡ every vertex of ¢¨£¥ §¦ either belongs to or is adjacent to some vertex in . -
High-Performance Parallel Graph Coloring with Strong Guarantees on Work, Depth, and Quality
High-Performance Parallel Graph Coloring with Strong Guarantees on Work, Depth, and Quality Maciej Besta1, Armon Carigiet1, Zur Vonarburg-Shmaria1, Kacper Janda2, Lukas Gianinazzi1, Torsten Hoefler1 1Department of Computer Science, ETH Zurich 2Faculty of Computer Science, Electronics and Telecommunications, AGH-UST Abstract—We develop the first parallel graph coloring heuris- to their degrees, random (R) [26] which chooses vertices tics with strong theoretical guarantees on work and depth and uniformly at random, incidence-degree (ID) [1] which picks coloring quality. The key idea is to design a relaxation of the vertices with the largest number of uncolored neighbors first, vertex degeneracy order, a well-known graph theory concept, and to color vertices in the order dictated by this relaxation. This saturation-degree (SD) [27], where a vertex whose neighbors introduces a tunable amount of parallelism into the degeneracy use the largest number of distinct colors is chosen first, and ordering that is otherwise hard to parallelize. This simple idea smallest-degree-last (SL) [28] that removes lowest degree enables significant benefits in several key aspects of graph color- vertices, recursively colors the resulting graph, and then colors ing. For example, one of our algorithms ensures polylogarithmic the removed vertices. All these ordering heuristics, combined depth and a bound on the number of used colors that is superior to all other parallelizable schemes, while maintaining work- with Greedy, have the inherent problem of no parallelism. efficiency. In addition to provable guarantees, the developed Jones and Plassmann combined this line of work with earlier algorithms have competitive run-times for several real-world parallel schemes for deriving maximum independent sets [29], graphs, while almost always providing superior coloring quality. -
Domatic Number of a Graph
European Journal of Molecular & Clinical Medicine ISSN 2515-8260 Volume 07, Issue 11, 2020 Domatic Number of a Graph M SUBHA1, Dr.T. Saravanan2 , T. Helan Magdalin3 1,2,3 New Prince Shri Bhavani College of Engineering and Technology, Chennai, India E-mail: [email protected] Abstract The domatic graph number is discussed in this chapter. Domatic was the word from the terms ‘Dominance’ and ‘Chromatic’ coined. Found domatic numbers for different type of graphs. 1. Introduction Graph Theory was developed in 1736 by Euler as a generalisation of the to the Koniberg bridges’ famous problem solution.. Graph dominance has been thoroughly researched in the graph theory division. Berge in 1958 and Ore in 1962 formalised dominance as a theoretical field in graph theory. Ore was characterized by minimal dominant sets. Historically, from chess came the first domination – type issues. Several chess players were interested in the minimum number of queens in the 1850s, so that any square on the chess board includes either a queen (remember that a queen can move any number of squares on the chess board horizontally, vertically, or diagonally). Definition: 4.1 A D-partition is a V(G) partition into a finite number of subsets. In every subset other than its own, each point in a subset is adjacent to at least one point in each subset. Definition: 4.2 A D-partition of G is a partition of V(G) into a finite number of subsets. Each point in a subset in any sub-set other than its own is adjacent to at least one point in each sub-set. -
The Domatic Number Problem: Boolean Hierarchy Completeness and Exact Exponential-Time Algorithms
The Domatic Number Problem: Boolean Hierarchy Completeness and Exact Exponential-Time Algorithms Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨at D¨usseldorf vorgelegt von Tobias Riege geboren in Haltern D¨usseldorf, im November 2006 Aus dem Institut f¨ur Informatik der Heinrich-Heine-Universit¨at D¨usseldorf Gedruckt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨at D¨usseldorf Referent: Prof. Dr. J¨org Rothe Koreferent: Prof. Dr. Egon Wanke Tag der m¨undlichen Pr¨ufung: 20.12.2006 iii Acknowledgments I want to thank my thesis advisor J¨org Rothe for all of his support during the past four years. In the first place, I am deeply grateful to him for giving me the chance to be part of his research team. Without his great efforts, I would never have had the chance to work in the scientific community. Many inspiring and valuable discussions with him initiated fruitful ideas that led to the research presented in this thesis. I am very thankful to him for letting me have a share of his mathematical precision. In addition, his profound knowledge of the English language helped a great deal to spice up our articles. Next I would like to thank all my coauthors, including J¨org Rothe, G´abor Erd´elyi, Holger Spakowski, and Masaki Yamamoto. Thanks also to Gerd Wechsung for pointing out that Exact-2-DNP is coNP-complete and Dieter Kratsch for giving the hint to Lawler’s algorithm for the colorability problem. Many thanks go out to all the great colleagues at the Department of Computer Science of the Heinrich-Heine-University D¨usseldorf.