DOCSLIB.ORG

SAT Approach for Decomposition Methods

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
SAT Approach for Decomposition Methods SAT Approach for Decomposition Methods DISSERTATION submitted in partial fulfillment of the requirements for the degree of Doktorin der Technischen Wissenschaften by M.Sc. Neha Lodha Registration Number 01428755 to the Faculty of Informatics at the TU Wien Advisor: Prof. Stefan Szeider Second advisor: Prof. Armin Biere The dissertation has been reviewed by: Daniel Le Berre Marijn J. H. Heule Vienna, 31st October, 2018 Neha Lodha Technische Universität Wien A-1040 Wien Karlsplatz 13 Tel. +43-1-58801-0 www.tuwien.ac.at Erklärung zur Verfassung der Arbeit M.Sc. Neha Lodha Favoritenstrasse 9, 1040 Wien Hiermit erkläre ich, dass ich diese Arbeit selbständig verfasst habe, dass ich die verwen- deten Quellen und Hilfsmittel vollständig angegeben habe und dass ich die Stellen der Arbeit – einschließlich Tabellen, Karten und Abbildungen –, die anderen Werken oder dem Internet im Wortlaut oder dem Sinn nach entnommen sind, auf jeden Fall unter Angabe der Quelle als Entlehnung kenntlich gemacht habe. Wien, 31. Oktober 2018 Neha Lodha iii Acknowledgements First and foremost, I would like to thank my advisor Prof. Stefan Szeider for guiding me through my journey as a Ph.D. student with his patience, constant motivation, and immense knowledge. His guidance helped me during the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor. Next, I would like to thank my co-advisor Prof. Armin Biere, my reviewers, Prof. Daniel Le Berre and Dr. Marijn Heule, and my Ph.D. committee, Prof. Reinhard Pichler and Prof. Florian Zuleger, for their insightful comments, encouragement, and patience. I am very thankful to Sebastian Ordyniak, who is my unofficial co-advisor, one of the closest collaborators, and somehow around him, all the bugs were automatically resolved. I am sincerely thankful to Prof. Matti Järvisalo and Prof. Torsten Schaub, who provided me with an opportunity to visit them and let me be a part of their research group. I gained crucial knowledge by collaborating with my co-authors, Simone Bova, Johannes Fichte, Robert Ganian, Ronald de Haan, and Markus Hecher. I am very fortunate to be a part of the doctoral program “LogiCS” at TU Wien funded by FWF, where I got to know world-class researchers from all around the world. I enjoyed the numerous fruitful discussions I have had with my peers; especially Eduard Eiben, Anna Lukina, Katalin Fazekas, Matthias Schlaipfer, Zeynep Gözen Saribatur, Fabian Klute, Tomáš Peitl, and Friedrich Slivovsky. Not to forget the invaluable administrative support I received from Anna Prianichnikova and our secretaries, Juliane Auerböck, Beatrix Bhul, Doris Dicklberger, Eva Nedoma, Karin Prater, and Ulrike Prohaska. I would like to express my gratitude towards Prof. Saket Saurabh, who recommended me to apply for this position and Prof. R. Ramanujan for encouraging me to venture out in the world. Last but not least, I would like to thank my family, my flatmates, and my friends for supporting me and providing me with a social life. I have had the great opportunity to be around a very active and collaborative environment during my Ph.D., I could not have wished for anything better. v Kurzfassung Wir suchen immer nach dem besten Weg ein Berechnungsproblem zu lösen. Jedoch kann der naheliegendste Weg, um ein gegebenes Problem zu lösen, manchmal nicht sehr praktisch sein. InformatikerInnen haben versucht praktische Wege zu finden, um schwere Probleme zu lösen. Eine der Haupttechnologien, die entwickelt wurden, um schwere Probleme zu lösen, sind strukturelle Zerlegungen. Diese ermöglichen eine gegebene Probleminstanz in kleine Teile zu zerlegen. Es gibt verschiedene Techniken die Zerlegungen zu benutzen, um ein Problem zu lösen, aber diese Techniken sind stark auf die Qualität der Zerlegung angewiesen. In dieser Arbeit entwickeln wir SAT-basierte Techniken, um „gute“ Zerlegungen zu finden. Satisfiablity (SAT) – auf Deutsch das Erfüllbarkeitsproblem der Aussagenlogik – ist eines der zentralsten Probleme in der Informatik. SAT-solver, das sind Werkzeuge, die dieses Problem lösen können, haben sich in den letzten Jahren drastisch verbessert. Aktuelle SAT-solver können einige mehrere MB große SAT-Instanzen innerhalb von Millisekunden lösen. Ein SAT-encoding ist die Transformation eines Berechnungsproblem nach SAT. Wir nutzen die Geschwindigkeit und Effizienz von SAT-basierten Techniken, um „gute“ Zerlegungen zu finden. In dieser Arbeit entwickeln wir SAT-encodings, um gute Zerlegungen zu finden. Eine der größten Hürden für SAT-encodings ist die Größe der SAT-Formel. Die meisten SAT-encodings sind zumindest kubisch in der Größe der Probleminstanz. Dies führt zu einem harten Limit für die Größe der zu lösenden Probleminstanz. Wir überwinden diese Restriktion indem wir eine neue SAT-basierte lokale Verbesserungstechnik entwickeln, in der wir versuchen, Teile einer gegebenen Zerlegungen mithilfe von SAT-encodings zu verbessern. Wir schlagen eine neue Charakterisierung für Branch-Zerlegungen vor und benutzen diese, um ein SAT-encoding für Branchwidth zu finden. Wir entwickeln einen lokalen Verbesserungsansatz für die Ermittlung besserer Branch-Zerlegungen, welche wir später auch für Baumzerlegungen benutzen. Wir entwickeln zwei verschiedene Charakterisierun- gen für spezielle Baumzerlegungen und Pfadzerlegungen und vergleichen diese empirisch. Zudem entwickeln wir auch ein SMT-encoding für die Ermittlung von fraktionalen Hyperbaum-Zerlegungen. vii Abstract We are always trying to find the best way to solve any computational problem. But, sometimes the most natural way to solve a given problem may not be very practical. Computer scientists have been trying to find practical ways for solving hard problems. One of the major technology, that was developed to solve hard problems, is decompositions. Decompositions allow one to decompose a given problem instance in small parts. There are various techniques that use decompositions to solve a problem, but these techniques rely heavily on the quality of the decompositions. In this thesis, we develop SAT-based techniques to find “good” decompositions. Satisfiability (SAT) is one of the most central problem in computer science and SAT- solvers, tools that can solve this problem, have improved drastically over the last years. Current SAT-solvers can solve some SAT instances, of several MB in size, in a matter of milliseconds. A SAT-encoding is the transformation of a computational problem to SAT. We make use of the speed and efficiency of the SAT based techniques for finding good decompositions. In this thesis we develop SAT-encodings for finding good decompositions. One of the major hurdle for SAT-encodings is the size of the SAT formula. Most of the SAT-encodings are at least cubic is the size of problem instance, which puts a hard limit on the size of the problem instance that can be solved. We overcome this restriction by developing a new SAT-based local improvement technique, where given a decomposition we try to improve parts of it using SAT-encodings. We propose a new characterization for branch decompositions and used it for a SAT- encoding for finding branchwidth. We develop local improvement approach for finding better branch decompositions, which we later used for tree decompositions as well. We develop two different characterizations for special tree decomposition and path decomposition, and compare them empirically. We also develop an SMT-encoding for finding fractional hypertree decompositions. ix Contents Kurzfassung vii Abstract ix Contents xi 1 Introduction 1 1.1 Decompositions . 1 1.2 SAT-Solvers . 3 1.3 Contribution . 4 1.4 Organization . 5 1.5 Software . 6 2 Preliminaries 9 2.1 Satisfiability . 9 2.2 Partitions . 11 2.3 Graphs . 11 2.4 Tree Decompositions . 14 2.5 Clique-width Decomposition . 18 2.6 Decompositions Addressed in this Thesis . 21 2.7 Experimental Setup and Benchmarks . 22 3 Overview of Results 25 3.1 Methodologies . 25 3.2 Results . 27 4 Branchwidth and Carving-Width 31 4.1 Introduction . 31 4.2 Preliminaries . 33 4.3 Tree Encoding for Branchwidth . 35 4.4 Partition-based Reformulation of Branchwidth . 38 4.5 Carving-Width . 43 4.6 Local Improvement for Branch Decompositions . 46 4.7 Local Improvement for Carving Decompositions . 52 xi 4.8 Experimental Results . 52 4.9 Chapter Summary . 58 5 Treewidth 59 5.1 Introduction . 59 5.2 Local Improvement of Tree Decompositions . 61 5.3 Experimental Results . 64 5.4 Chapter Summary . 66 6 Special Treewidth and Pathwidth 69 6.1 Motivation . 69 6.2 Preliminaries . 72 6.3 Preprocessing . 75 6.4 Partition-Based Approach for Special Treewidth . 86 6.5 Ordering-Based Approach for Special Treewidth . 91 6.6 SAT-Encodings for Pathwidth . 96 6.7 Experiments . 99 6.8 Chapter Summary . 109 7 Fractional Hypertree Width 111 7.1 Introduction . 111 7.2 Preliminaries . 113 7.3 Ordering-Based Characterization of Fractional Hypertree Width . 114 7.4 SMT-encoding for Fractional Hypertree Decomposition . 117 7.5 Preprocessing . 119 7.6 Symmetry Breaking and Lower Bounds with Cliques . 122 7.7 Experimental Work . 123 7.8 Chapter Summary . 128 8 Conclusion and Future Work 131 List of Figures 133 List of Tables 135 List of Algorithms 137 Bibliography 139 CHAPTER 1 Introduction Computer science has become an essential part of almost every technology; querying a database, verifying software and hardware, scheduling processes or events are a few to name. As it turns out, many of these problems are not easy to solve. In case of some of these problems, we expect that one can only verify a given solution efficiently. Fortunately, there exists a silver lining for this predicament. Namely, in some special cases, it is not necessary to solve the entire problem instance. Instead, we can find a solution by first splitting the problem instance in parts, solving these and later combining the partial solutions. The splitting of a problem instance in parts is called “decomposition” and there exist various techniques to achieve this. Each decomposition method has some kind of “width measure” associated with it, which represents the quality of a decomposition.
Load more

Recommended publications
  • The Unit Distance Graph Problem and Equivalence Relations
    Math/CCS 103 Professor: Padraic Bartlett Lecture 11: The Unit Distance Graph Week 8 UCSB 2014 (Source: \The Mathematical Coloring Book," by Alexander Soifer.) 1 The Unit Distance Graph Problem 2 Definition. Consider the following method for turning R into a graph: 2 • Vertices: all points in R . • Edges: connect any two points (a; b) and (c; d) iff the distance between them is exactly 1. This graph is called the unit distance graph. Visualizing this is kinda tricky | it's got an absolutely insane number of vertices and edges. However, we can ask a question about it: Question. How many colors do we need in order to create a proper coloring of the unit distance graph? So: the answer isn't immediately obvious (right?) Instead, what we're going to try to do is just bound the possible answers, to get an idea of what the answers might be. How can we even bound such a thing? Well: to get a lower bound, it suffices to consider finite graphs G that we can draw in the plane using only straight edges of length 1. Because 2 our graph on R must contain any such graph \inside" of itself, examining these graphs will give us some easy lower bounds! So, by examining a equilateral triangle T , which has χ(T ) = 3, we can see that 2 χ(R ) ≥ 3: This is because it takes three colors to color an equilateral triangle's vertices in such a way that no edge has two endpoints of the same color. Similarly, by examining the following pentagonal construction (called a Moser spindle,) 1 we can actually do one better and say that 2 χ(R ) ≥ 4: Verify for yourself that you can't color this graph with three colors! 2 Conversely: to exhibit an upper bound on χ(R ) of k, it suffices to create a way of \painting" the plane with k-colors in such a way that no two points distance 1 apart get the same color.
    [Show full text]
  • Snarks and Flow-Critical Graphs 1
    Snarks and Flow-Critical Graphs 1 CˆandidaNunes da Silva a Lissa Pesci a Cl´audioL. Lucchesi b a DComp – CCTS – ufscar – Sorocaba, sp, Brazil b Faculty of Computing – facom-ufms – Campo Grande, ms, Brazil Abstract It is well-known that a 2-edge-connected cubic graph has a 3-edge-colouring if and only if it has a 4-flow. Snarks are usually regarded to be, in some sense, the minimal cubic graphs without a 3-edge-colouring. We defined the notion of 4-flow-critical graphs as an alternative concept towards minimal graphs. It turns out that every snark has a 4-flow-critical snark as a minor. We verify, surprisingly, that less than 5% of the snarks with up to 28 vertices are 4-flow-critical. On the other hand, there are infinitely many 4-flow-critical snarks, as every flower-snark is 4-flow-critical. These observations give some insight into a new research approach regarding Tutte’s Flow Conjectures. Keywords: nowhere-zero k-flows, Tutte’s Flow Conjectures, 3-edge-colouring, flow-critical graphs. 1 Nowhere-Zero Flows Let k > 1 be an integer, let G be a graph, let D be an orientation of G and let ϕ be a weight function that associates to each edge of G a positive integer in the set {1, 2, . , k − 1}. The pair (D, ϕ) is a (nowhere-zero) k-flow of G 1 Support by fapesp, capes and cnpq if every vertex v of G is balanced, i. e., the sum of the weights of all edges leaving v equals the sum of the weights of all edges entering v.
    [Show full text]
  • Eindhoven University of Technology BACHELOR on the K-Independent
    Eindhoven University of Technology BACHELOR On the k-Independent Set Problem Koerts, Hidde O. Award date: 2021 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain On the k-Independent Set Problem Hidde Koerts Supervised by Aida Abiad February 1, 2021 Hidde Koerts Supervised by Aida Abiad Abstract In this thesis, we study several open problems related to the k-independence num- ber, which is defined as the maximum size of a set of vertices at pairwise dis- tance greater than k (or alternatively, as the independence number of the k-th graph power). Firstly, we extend the definitions of vertex covers and cliques to allow for natural extensions of the equivalencies between independent sets, ver- tex covers, and cliques.
    [Show full text]
  • A Proof of Tomescu's Graph Coloring Conjecture
    A proof of Tomescu’s graph coloring conjecture Jacob Fox,∗ Xiaoyu He,† Freddie Manners‡ October 23, 2018 Abstract In 1971, Tomescu conjectured that every connected graph G on n vertices with chromatic number n−k k ≥ 4 has at most k!(k − 1) proper k-colorings. Recently, Knox and Mohar proved Tomescu’s conjecture for k = 4 and k = 5. In this paper, we complete the proof of Tomescu’s conjecture for all k ≥ 4, and show that equality occurs if and only if G is a k-clique with trees attached to each vertex. 1 Introduction Let k be a positive integer and G = (V, E) be a graph1.A proper k-coloring, or simply a k-coloring, of G is a function c : V → [k] (here [k] = {1, . , k}) such that c(u) 6= c(v) whenever uv ∈ E. The chromatic number χ(G) is the minimum k for which there exists a k-coloring of G. Let PG(k) denote the number of k-colorings of G. This function is a polynomial in k and is thus called the chromatic polynomial of G. In 1912, Birkhoff [5] introduced the chromatic polynomial for planar graphs in an attempt to solve the Four Color Problem using tools from analysis. The chromatic polynomial was later defined and studied for general graphs by Whitney [41]. Despite a great deal of attention over the past century, our understanding of the chromatic polynomial is still quite poor. In particular, proving general bounds on the chromatic polynomial remains a major challenge. The first result of this type, due to Birkhoff [6], states that n−3 PG(k) ≥ k(k − 1)(k − 2)(k − 3) for any planar graph G on n vertices and any real number k ≥ 5.
    [Show full text]
  • Toward a Unit Distance Embedding for the Heawood Graph
    Toward a Unit Distance Embedding for the Heawood graph Mitchell A. Harris1 Harvard Medical School Department of Radiology Boston, MA 02114, USA [email protected] Abstract. The unit distance embeddability of a graph, like planarity, involves a mix of constraints that are combinatorial and geometric. We construct a unit distance embedding for H − e in the hope that it will lead to an embedding for H. We then investigate analytical methods for a general decision procedure for testing unit distance embeddability. 1 Introduction Unit distance embedding of a graph is an assignment of coordinates in the plane to the vertices of a graph such that if there is an edge between two vertices, then their respective coordinates are exactly distance 1 apart. To bar trivial embed- dings, such as for bipartite graphs having all nodes in one part located at (0, 0), for the other part at (0, 1), it is also required that the embedded points be dis- tinct. There is no restriction on edge crossing. A graph is said to be unit distance embeddable if there exists such an embedding (with the obvious abbreviations arXiv:0711.1157v1 [math.CO] 7 Nov 2007 employed, such as UD embeddable, a UDG, etc). For some well-known examples, K4 is not UDG but K4 e is. The Moser spindle is UDG, and is also 4 colorable [5], giving the largest− currently known lower bound to the Erd¨os colorability of the plane [3]. The graph K2,3 is not be- cause from two given points in the plane there are exactly two points of distance 1, but the graph wants three; removing any edge allows a UD embedding.
    [Show full text]
  • Generation of Cubic Graphs and Snarks with Large Girth
    Generation of cubic graphs and snarks with large girth Gunnar Brinkmanna, Jan Goedgebeura,1 aDepartment of Applied Mathematics, Computer Science & Statistics Ghent University Krijgslaan 281-S9, 9000 Ghent, Belgium Abstract We describe two new algorithms for the generation of all non-isomorphic cubic graphs with girth at least k ≥ 5 which are very efficient for 5 ≤ k ≤ 7 and show how these algorithms can be efficiently restricted to generate snarks with girth at least k. Our implementation of these algorithms is more than 30, respectively 40 times faster than the previously fastest generator for cubic graphs with girth at least 6 and 7, respec- tively. Using these generators we have also generated all non-isomorphic snarks with girth at least 6 up to 38 vertices and show that there are no snarks with girth at least 7 up to 42 vertices. We present and analyse the new list of snarks with girth 6. Keywords: cubic graph, snark, girth, chromatic index, exhaustive generation 1. Introduction A cubic (or 3-regular) graph is a graph where every vertex has degree 3. Cubic graphs have interesting applications in chemistry as they can be used to represent molecules (where the vertices are e.g. carbon atoms such as in fullerenes [20]). Cubic graphs are also especially interesting in mathematics since for many open problems in graph theory it has been proven that cubic graphs are the smallest possible potential counterexamples, i.e. that if the conjecture is false, the smallest counterexample must be a cubic graph. For examples, see [5]. For most problems the possible counterexamples can be further restricted to the sub- class of cubic graphs which are not 3-edge-colourable.
    [Show full text]
  • Pebbling in Watkins Snark Graph
    International Journal of Research in Advent Technology, Vol.7, No.2, March 2019 E-ISSN: 2321-9637 Available online at www.ijrat.org Pebbling In Watkins Snark Graph Ms.Sreedevi S1, Dr.M.S Anilkumar2, Department of Mathematics, Mahatma Gandhi College, Thiruvananthapuram, Kerala1, 2 Email:[email protected],[email protected] Abstract- The pebbling theory is a study of a mathematical game that is played over a graph. In this game, a pebbling movement means removing two pebbles from a given vertex and adding one pebble to one of its neighbors and removing the other pebble from the game. The pebbling number of a graph G is defined a smallest positive integer required to add a pebble at any target vertex of the graph. It is denoted as . Every vertex of the graph is pebbled irrespective of the initial pattern of pebbles. Cubic graph is also called a 3- regular graph which is used in a real time scenario. In this paper, we have determined the pebbling number of Watkins Snark by constructing a Watkins Flower Snark of vertices, edges, cycles and disjoint sets which are present in the Watkins Snark. It is a connected graph in which bridgeless cubic index is equal to 4 with 75 edges and 50 vertices. Keywords - Graphs; pebbling number; Watkins Snark graph 1. INTRODUCTION 1.3 Solvability Throughout this paper, let denotes a simple connected graph. n and are the number Consider Peterson graph with pebbles on the vertices. of vertices and edges respectively in G and the diameter From the (Figure-1), one can move 2 pebbles among 3 of G is denoted as d.
    [Show full text]
  • On Hypohamiltonian Snarks and a Theorem of Fiorini
    On Hypohamiltonian Snarks and a Theorem of Fiorini Jan Goedgebeur ∗ Department of Applied Mathematics, Computer Science & Statistics, Ghent University, Krijgslaan 281-S9, 9000 Ghent, Belgium Carol T. Zamfirescu † Department of Applied Mathematics, Computer Science & Statistics, Ghent University, Krijgslaan 281-S9, 9000 Ghent, Belgium In memory of the first author’s little girl Ella. Received dd mmmm yyyy, accepted dd mmmmm yyyy, published online dd mmmmm yyyy Abstract We discuss an omission in the statement and proof of Fiorini’s 1983 theorem on hypo- hamiltonian snarks and present a version of this theorem which is more general in several ways. Using Fiorini’s erroneous result, Steffen showed that hypohamiltonian snarks exist for some n ≥ 10 and each even n ≥ 92. We rectify Steffen’s proof by providing a correct demonstration of a technical lemma on flower snarks, which might be of separate interest. We then strengthen Steffen’s theorem to the strongest possible form by determining all or- ders for which hypohamiltonian snarks exists. This also strengthens a result of M´aˇcajov´a and Skoviera.ˇ Finally, we verify a conjecture of Steffen on hypohamiltonian snarks up to 36 vertices. Keywords: hypohamiltonian, snark, irreducible snark, dot product Math. Subj. Class.: 05C10, 05C38, 05C45, 05C85. arXiv:1608.07164v1 [math.CO] 25 Aug 2016 1 Introduction A graph G is hypohamiltonian if G itself is non-hamiltonian, but for every vertex v in G, the graph G−v is hamiltonian. A snark shall be a cubic cyclically 4-edge-connected graph with chromatic index 4 (i.e. four colours are required in any proper edge-colouring) and girth at least 5.
    [Show full text]
  • Beatrice Ruini
    LE MATEMATICHE Vol. LXV (2010) – Fasc. I, pp. 3–21 doi: 10.4418/2010.65.1.1 SOME INFINITE CLASSES OF ASYMMETRIC NEARLY HAMILTONIAN SNARKS CARLA FIORI - BEATRICE RUINI We determine the full automorphism group of each member of three infinite families of connected cubic graphs which are snarks. A graph is said to be nearly hamiltonian if it has a cycle which contains all vertices but one. We prove, in particular, that for every possible order n ≥ 28 there exists a nearly hamiltonian snark of order n with trivial automorphism group. 1. Introduction Snarks are non-trivial connected cubic graphs which do not admit a 3-edge- coloring (a precise definition will be given below). The term snark owes its origin to Lewis Carroll’s famouse nonsense poem “The Hunting of the Snark”. It was introduced as a graph theoretical term by Gardner in [13] when snarks were thought to be very rare and unusual “creatures”. Tait initiated the study of snarks in 1880 when he proved that the Four Color Theorem is equivalent to the statement that no snark is planar. Asymmetric graphs are graphs pos- sessing a single graph automorphism -the identity- and for that reason they are also called identity graphs. Twenty-seven examples of asymmetric graphs are illustrated in [27]. Two of them are the snarks Sn8 and Sn9 of order 20 listed in [21] p. 276. Asymmetric graphs have been the subject of many studies, see, Entrato in redazione: 4 febbraio 2009 AMS 2000 Subject Classification: 05C15, 20B25. Keywords: full automorphism group, snark, nearly hamiltonian snark.
    [Show full text]
  • SNARKS Generation, Coverings and Colourings
    SNARKS Generation, coverings and colourings jonas hägglund Doctoral thesis Department of mathematics and mathematical statistics Umeå University April 2012 Doctoral dissertation Department of mathematics and mathematical statistics Umeå University SE-901 87 Umeå Sweden Copyright © 2012 Jonas Hägglund Doctoral thesis No. 53 ISBN: 978-91-7459-399-0 ISSN: 1102-8300 Printed by Print & Media Umeå 2012 To my family. ABSTRACT For a number of unsolved problems in graph theory such as the cycle double cover conjecture, Fulkerson’s conjecture and Tutte’s 5-flow conjecture it is sufficient to prove them for a family of graphs called snarks. Named after the mysterious creature in Lewis Carroll’s poem, a snark is a cyclically 4-edge connected 3-regular graph of girth at least 5 which cannot be properly edge coloured using three colours. Snarks and problems for which an edge minimal counterexample must be a snark are the central topics of this thesis. The first part of this thesis is intended as a short introduction to the area. The second part is an introduction to the appended papers and the third part consists of the four papers presented in a chronological order. In Paper I we study the strong cycle double cover conjecture and stable cycles for small snarks. We prove that if a bridgeless cubic graph G has a cycle of length at least jV(G)j - 9 then it also has a cycle double cover. Furthermore we show that there exist cyclically 5-edge connected snarks with stable cycles and that there exists an infinite family of snarks with stable cycles of length 24.
    [Show full text]
  • Graph Theory Graph Theory (II)
    J.A. Bondy U.S.R. Murty Graph Theory (II) ABC J.A. Bondy, PhD U.S.R. Murty, PhD Universite´ Claude-Bernard Lyon 1 Mathematics Faculty Domaine de Gerland University of Waterloo 50 Avenue Tony Garnier 200 University Avenue West 69366 Lyon Cedex 07 Waterloo, Ontario, Canada France N2L 3G1 Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA Graduate Texts in Mathematics series ISSN: 0072-5285 ISBN: 978-1-84628-969-9 e-ISBN: 978-1-84628-970-5 DOI: 10.1007/978-1-84628-970-5 Library of Congress Control Number: 2007940370 Mathematics Subject Classification (2000): 05C; 68R10 °c J.A. Bondy & U.S.R. Murty 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or trans- mitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered name, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.
    [Show full text]
  • On Measures of Edge-Uncolorability of Cubic Graphs: a Brief Survey and Some New Results
    On measures of edge-uncolorability of cubic graphs: A brief survey and some new results M.A. Fiola, G. Mazzuoccolob, E. Steffenc aBarcelona Graduate School of Mathematics and Departament de Matem`atiques Universitat Polit`ecnicade Catalunya Jordi Girona 1-3 , M`odulC3, Campus Nord 08034 Barcelona, Catalonia. bDipartimento di Informatica Universit´adi Verona Strada le Grazie 15, 37134 Verona, Italy. cPaderborn Center for Advanced Studies and Institute for Mathematics Universit¨atPaderborn F¨urstenallee11, D-33102 Paderborn, Germany. E-mails: [email protected], [email protected], [email protected] February 24, 2017 Abstract There are many hard conjectures in graph theory, like Tutte's 5-flow conjec- arXiv:1702.07156v1 [math.CO] 23 Feb 2017 ture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for 3-edge-colorable cubic graphs, cubic graphs which are not 3-edge-colorable, often called snarks, play a key role in this context. Here, we survey parameters measuring how far apart a non 3-edge-colorable graph is from being 3-edge- colorable. We study their interrelation and prove some new results. Besides getting new insight into the structure of snarks, we show that such measures give partial results with respect to these important conjectures. The paper closes with a list of open problems and conjectures. 1 Mathematics Subject Classifications: 05C15, 05C21, 05C70, 05C75. Keywords: Cubic graph; Tait coloring; snark; Boole coloring; Berge's conjecture; Tutte's 5-flow conjecture; Fulkerson's Conjecture.
    [Show full text]