The Edge Grundy Numbers of Some Graphs
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A Fast and Scalable Graph Coloring Algorithm for Multi-Core and Many-Core Architectures
A Fast and Scalable Graph Coloring Algorithm for Multi-core and Many-core Architectures Georgios Rokos1, Gerard Gorman2, and Paul H J Kelly1 1 Software Peroformance Optimisation Group, Department of Computing 2 Applied Modelling and Computation Group, Department of Earth Science and Engineering Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom, fgeorgios.rokos09,g.gorman,[email protected] Abstract. Irregular computations on unstructured data are an impor- tant class of problems for parallel programming. Graph coloring is often an important preprocessing step, e.g. as a way to perform dependency analysis for safe parallel execution. The total run time of a coloring algo- rithm adds to the overall parallel overhead of the application whereas the number of colors used determines the amount of exposed parallelism. A fast and scalable coloring algorithm using as few colors as possible is vi- tal for the overall parallel performance and scalability of many irregular applications that depend upon runtime dependency analysis. C¸ataly¨ureket al. have proposed a graph coloring algorithm which relies on speculative, local assignment of colors. In this paper we present an improved version which runs even more optimistically with less thread synchronization and reduced number of conflicts compared to C¸ataly¨urek et al.'s algorithm. We show that the new technique scales better on multi- core and many-core systems and performs up to 1.5x faster than its pre- decessor on graphs with high-degree vertices, while keeping the number of colors at the same near-optimal levels. Keywords: Graph Coloring; Greedy Coloring; First-Fit Coloring; Ir- regular Data; Parallel Graph Algorithms; Shared-Memory Parallelism; Optimistic Execution; Many-core Architectures; Intel®Xeon Phi™ 1 Introduction Many modern applications are built around algorithms which operate on irreg- ular data structures, usually in form of graphs. -
I?'!'-Comparability Graphs
Discrete Mathematics 74 (1989) 173-200 173 North-Holland I?‘!‘-COMPARABILITYGRAPHS C.T. HOANG Department of Computer Science, Rutgers University, New Brunswick, NJ 08903, U.S.A. and B.A. REED Discrete Mathematics Group, Bell Communications Research, Morristown, NJ 07950, U.S.A. In 1981, Chv&tal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs. In this paper, we introduce a new class of perfectly orderable graphs, the &comparability graphs. This class generalizes comparability graphs in a natural way. We also prove a decomposition theorem which leads to a structural characteriza- tion of &comparability graphs. Using this characterization, we develop a polynomial-time recognition algorithm and polynomial-time algorithms for the clique and colouring problems for &comparability graphs. 1. Introduction A natural way to color the vertices of a graph is to put them in some order v*cv~<” l < v, and then assign colors in the following manner: (1) Consider the vertxes sequentially (in the sequence given by the order) (2) Assign to each Vi the smallest color (positive integer) not used on any neighbor vi of Vi with j c i. We shall call this the greedy coloring algorithm. An order < of a graph G is perfect if for each subgraph H of G, the greedy algorithm applied to H, gives an optimal coloring of H. An obstruction in a ordered graph (G, <) is a set of four vertices {a, b, c, d} with edges ab, bc, cd (and no other edge) and a c b, d CC. Clearly, if < is a perfect order on G, then (G, C) contains no obstruction (because the chromatic number of an obstruction is twu, but the greedy coloring algorithm uses three colors). -
The Hadwiger Number, Chordal Graphs and Ab-Perfection Arxiv
The Hadwiger number, chordal graphs and ab-perfection∗ Christian Rubio-Montiel [email protected] Instituto de Matem´aticas, Universidad Nacional Aut´onomade M´exico, 04510, Mexico City, Mexico Department of Algebra, Comenius University, 84248, Bratislava, Slovakia October 2, 2018 Abstract A graph is chordal if every induced cycle has three vertices. The Hadwiger number is the order of the largest complete minor of a graph. We characterize the chordal graphs in terms of the Hadwiger number and we also characterize the families of graphs such that for each induced subgraph H, (1) the Hadwiger number of H is equal to the maximum clique order of H, (2) the Hadwiger number of H is equal to the achromatic number of H, (3) the b-chromatic number is equal to the pseudoachromatic number, (4) the pseudo-b-chromatic number is equal to the pseudoachromatic number, (5) the arXiv:1701.08417v1 [math.CO] 29 Jan 2017 Hadwiger number of H is equal to the Grundy number of H, and (6) the b-chromatic number is equal to the pseudo-Grundy number. Keywords: Complete colorings, perfect graphs, forbidden graphs characterization. 2010 Mathematics Subject Classification: 05C17; 05C15; 05C83. ∗Research partially supported by CONACyT-Mexico, Grants 178395, 166306; PAPIIT-Mexico, Grant IN104915; a Postdoctoral fellowship of CONACyT-Mexico; and the National scholarship programme of the Slovak republic. 1 1 Introduction Let G be a finite graph. A k-coloring of G is a surjective function & that assigns a number from the set [k] := 1; : : : ; k to each vertex of G.A k-coloring & of G is called proper if any two adjacent verticesf haveg different colors, and & is called complete if for each pair of different colors i; j [k] there exists an edge xy E(G) such that x &−1(i) and y &−1(j). -
Building a Larger Class of Graphs for Efficient Reconfiguration
Building a larger class of graphs for efficient reconfiguration of vertex colouring by Owen Merkel A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, Canada, 2020 c Owen Merkel 2020 This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Statement of Contributions Owen Merkel is the sole author of Chapters 1, 2, 3, and 6 which were written under the supervision of Prof. Therese Biedl and Prof. Anna Lubiw and were not written for publication. Exceptions to sole authorship are Chapters 4 and 5, which is joint work with Prof. Therese Biedl and Prof. Anna Lubiw and consists in part of a manuscript written for publication. iii Abstract A k-colouring of a graph G is an assignment of at most k colours to the vertices of G so that adjacent vertices are assigned different colours. The reconfiguration graph of the k-colourings, Rk(G), is the graph whose vertices are the k-colourings of G and two colourings are joined by an edge in Rk(G) if they differ in colour on exactly one vertex. For a k-colourable graph G, we investigate the connectivity and diameter of Rk+1(G). It is known that not all weakly chordal graphs have the property that Rk+1(G) is connected. -
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. -
Graph Coloring Major Themes Vertex Colorings Χ(G) =
CSC 1300 – Discrete Structures 17: Graph Coloring Major Themes • Vertex coloring • Chromac number χ(G) Graph Coloring • Map coloring • Greedy coloring algorithm • CSC 1300 – Discrete Structures Applicaons Villanova University Villanova CSC 1300 - Dr Papalaskari 1 Villanova CSC 1300 - Dr Papalaskari 2 What is the least number of colors needed for the verFces of Vertex Colorings this graph so that no two adjacent verFces have the same color? Adjacent ver,ces cannot have the same color χ(G) = Chromac number χ(G) = least number of colors needed to color the verFces of a graph so that no two adjacent verFces are assigned the same color? Source: “Discrete Mathemacs” by Chartrand & Zhang, 2011, Waveland Press. Source: “Discrete Mathemacs with Ducks” by Sara-Marie Belcastro, 2012, CRC Press, Fig 13.1. 4 5 Dr Papalaskari 1 CSC 1300 – Discrete Structures 17: Graph Coloring Map Coloring Map Coloring What is the least number of colors needed to color a map? Region è vertex Common border è edge G A D C E B F G H I Villanova CSC 1300 - Dr Papalaskari 8 Villanova CSC 1300 - Dr Papalaskari 9 Coloring the USA Four color theorem Every planar graph is 4-colorable The proof of this theorem is one of the most famous and controversial proofs in mathemacs, because it relies on a computer program. It was first presented in 1976. A more recent reformulaon can be found in this arFcle: Formal Proof – The Four Color Theorem, Georges Gonthier, NoFces of the American Mathemacal Society, December 2008. hp://www.printco.com/pages/State%20Map%20Requirements/USA-colored-12-x-8.gif -
Results on the Grundy Chromatic Number of Graphs Manouchehr Zaker
Discrete Mathematics 306 (2006) 3166–3173 www.elsevier.com/locate/disc Results on the Grundy chromatic number of graphs Manouchehr Zaker Institute for Advanced Studies in Basic Sciences, 45195-1159 Zanjan, Iran Received 7 November 2003; received in revised form 22 June 2005; accepted 22 June 2005 Available online 7 August 2006 Abstract Given a graph G,byaGrundy k-coloring of G we mean any proper k-vertex coloring of G such that for each two colors i and j, i<j, every vertex of G colored by j has a neighbor with color i. The maximum k for which there exists a Grundy k-coloring is denoted by (G) and called Grundy (chromatic) number of G. We first discuss the fixed-parameter complexity of determining (G)k, for any fixed integer k and show that it is a polynomial time problem. But in general, Grundy number is an NP-complete problem. We show that it is NP-complete even for the complement of bipartite graphs and describe the Grundy number of these graphs in terms of the minimum edge dominating number of their complements. Next we obtain some additive Nordhaus–Gaddum-type inequalities concerning (G) and (Gc), for a few family of graphs. We introduce well-colored graphs, which are graphs G for which applying every greedy coloring results in a coloring of G with (G) colors. Equivalently G is well colored if (G) = (G). We prove that the recognition problem of well-colored graphs is a coNP-complete problem. © 2006 Elsevier B.V. All rights reserved. Keywords: Colorings; Chromatic number; Grundy number; First-fit colorings; NP-complete; Edge dominating sets 1. -
Three Years of Graphs and Music: Some Results in Graph Theory and Its
Three years of graphs and music : some results in graph theory and its applications Nathann Cohen To cite this version: Nathann Cohen. Three years of graphs and music : some results in graph theory and its applications. Discrete Mathematics [cs.DM]. Université Nice Sophia Antipolis, 2011. English. tel-00645151 HAL Id: tel-00645151 https://tel.archives-ouvertes.fr/tel-00645151 Submitted on 26 Nov 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITE´ DE NICE-SOPHIA ANTIPOLIS - UFR SCIENCES ECOLE´ DOCTORALE STIC SCIENCES ET TECHNOLOGIES DE L’INFORMATION ET DE LA COMMUNICATION TH ESE` pour obtenir le titre de Docteur en Sciences de l’Universite´ de Nice - Sophia Antipolis Mention : INFORMATIQUE Present´ ee´ et soutenue par Nathann COHEN Three years of graphs and music Some results in graph theory and its applications These` dirigee´ par Fred´ eric´ HAVET prepar´ ee´ dans le Projet MASCOTTE, I3S (CNRS/UNS)-INRIA soutenue le 20 octobre 2011 Rapporteurs : Jørgen BANG-JENSEN - Professor University of Southern Denmark, Odense Daniel KRAL´ ’ - Associate -
Algorithmic Graph Theory Part III Perfect Graphs and Their Subclasses
Algorithmic Graph Theory Part III Perfect Graphs and Their Subclasses Martin Milanicˇ [email protected] University of Primorska, Koper, Slovenia Dipartimento di Informatica Universita` degli Studi di Verona, March 2013 1/55 What we’ll do 1 THE BASICS. 2 PERFECT GRAPHS. 3 COGRAPHS. 4 CHORDAL GRAPHS. 5 SPLIT GRAPHS. 6 THRESHOLD GRAPHS. 7 INTERVAL GRAPHS. 2/55 THE BASICS. 2/55 Induced Subgraphs Recall: Definition Given two graphs G = (V , E) and G′ = (V ′, E ′), we say that G is an induced subgraph of G′ if V ⊆ V ′ and E = {uv ∈ E ′ : u, v ∈ V }. Equivalently: G can be obtained from G′ by deleting vertices. Notation: G < G′ 3/55 Hereditary Graph Properties Hereditary graph property (hereditary graph class) = a class of graphs closed under deletion of vertices = a class of graphs closed under taking induced subgraphs Formally: a set of graphs X such that G ∈ X and H < G ⇒ H ∈ X . 4/55 Hereditary Graph Properties Hereditary graph property (Hereditary graph class) = a class of graphs closed under deletion of vertices = a class of graphs closed under taking induced subgraphs Examples: forests complete graphs line graphs bipartite graphs planar graphs graphs of degree at most ∆ triangle-free graphs perfect graphs 5/55 Hereditary Graph Properties Why hereditary graph classes? Vertex deletions are very useful for developing algorithms for various graph optimization problems. Every hereditary graph property can be described in terms of forbidden induced subgraphs. 6/55 Hereditary Graph Properties H-free graph = a graph that does not contain H as an induced subgraph Free(H) = the class of H-free graphs Free(M) := H∈M Free(H) M-free graphT = a graph in Free(M) Proposition X hereditary ⇐⇒ X = Free(M) for some M M = {all (minimal) graphs not in X} The set M is the set of forbidden induced subgraphs for X. -
Greedy Defining Sets of Graphs
Greedy defining sets of graphs Manouchehr Zaker Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11365-9415, Tehran, IRAN [email protected] Abstract For a graph G and an order a on V(G), we define a greedy defining set as a subset S of V(G) with an assignment of colors to vertices in S, such that the pre-coloring can be extended to a x( G)-coloring of G by the greedy coloring of (G, a). A greedy defining set ofaX( G)-coloring C of G is a greedy defining set, which results in the coloring C (by the greedy procedure). We denote the size of a greedy defining set of C with minimum cardinality by G D N (G, a, C). In this paper we show that the problem of determining GDN(G,a,C), for an instance (G,a,C) is an NP-complete problem. 1 Introduction and preliminaries We begin with some terminology and notation which are used throughout the paper. For the other necessary definitions and notation, we refer the reader to texts such as [4]. A harmonious coloring of a graph G is a proper vertex coloring of G, such that the subgraph induced on any two different colors has at most one edge. By a hypergraph we mean an ordinary hypergraph which consists of a vertex set V and a collection of subsets of V, called (hyper)edges. In a hypergraph 1£ a block ing set is a subset of the vertex set which intersects every edge of 1£. -
Chordal Graphs MPRI 2017–2018
Chordal graphs MPRI 2017{2018 Chordal graphs MPRI 2017{2018 Michel Habib [email protected] http://www.irif.fr/~habib Sophie Germain, septembre 2017 Chordal graphs MPRI 2017{2018 Schedule Chordal graphs Representation of chordal graphs LBFS and chordal graphs More structural insights of chordal graphs Other classical graph searches and chordal graphs Greedy colorings Chordal graphs MPRI 2017{2018 Chordal graphs Definition A graph is chordal iff it has no chordless cycle of length ≥ 4 or equivalently it has no induced cycle of length ≥ 4. I Chordal graphs are hereditary I Interval graphs are chordal Chordal graphs MPRI 2017{2018 Chordal graphs Applications I Many NP-complete problems for general graphs are polynomial for chordal graphs. I Graph theory : Treewidth (resp. pathwidth) are very important graph parameters that measure distance from a chordal graph (resp. interval graph). 1 I Perfect phylogeny 1. In fact chordal graphs were first defined in a biological modelisation pers- pective ! Chordal graphs MPRI 2017{2018 Chordal graphs G.A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg, 38 (1961), pp. 71{76 Chordal graphs MPRI 2017{2018 Representation of chordal graphs About Representations I Interval graphs are chordal graphs I How can we represent chordal graphs ? I As an intersection of some family ? I This family must generalize intervals on a line Chordal graphs MPRI 2017{2018 Representation of chordal graphs Fundamental objects to play with I Maximal Cliques under inclusion I We can have exponentially many maximal cliques : A complete graph Kn in which each vertex is replaced by a pair of false twins has exactly 2n maximal cliques. -
Graph Theory, Part 2
Graph Theory, Part 2 7 Coloring Suppose that you are responsible for scheduling times for lectures in a university. You want to make sure that any two lectures with a common student occur at different times to avoid a conflict. We could put the various lectures on a chart and mark with an \X" any pair that has students in common: Lecture A C G H I L M P S Astronomy X X X X Chemistry X X Greek X X X X X History X X X X Italian X X X Latin X X X X X X Music X X X X Philosophy X X Spanish X X X X A more convenient representation of this information is a graph with one vertex for each lecture and in which two vertices are joined if there is a conflict between them: S H P C I L G A M Now, we cannot schedule two lectures at the same time if there is a conflict, but we would like to use as few separate times as possible, subject to this constraint. How many different times are necessary? We can code each time with a color, for example 11:00-12:00 might be given the color green, and those lectures that meet at this time will be colored green. The no-conflict rule then means that we need to color the vertices of our graph in such a way that no two adjacent vertices (representing courses which conflict with each other) have the same color. This motivates us to make a definition: Definition 15 (Proper Coloring, k-Coloring, k-Colorable).