Graph Theory Graph Theory (II)
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Directed Cycle Double Cover Conjecture: Fork Graphs
Directed Cycle Double Cover Conjecture: Fork Graphs Andrea Jim´enez ∗ Martin Loebl y October 22, 2013 Abstract We explore the well-known Jaeger's directed cycle double cover conjecture which is equiva- lent to the assertion that every cubic bridgeless graph has an embedding on a closed orientable surface with no dual loop. We associate each cubic graph G with a novel object H that we call a hexagon graph; perfect matchings of H describe all embeddings of G on closed orientable surfaces. The study of hexagon graphs leads us to define a new class of graphs that we call lean fork-graphs. Fork graphs are cubic bridgeless graphs obtained from a triangle by sequentially connecting fork-type graphs and performing Y−∆, ∆−Y transformations; lean fork-graphs are fork graphs fulfilling a connectivity property. We prove that Jaeger's conjecture holds for the class of lean fork-graphs. The class of lean fork-graphs is rich; namely, for each cubic bridgeless graph G there is a lean fork-graph containing a subdivision of G as an induced subgraph. Our results establish for the first time, to the best of our knowledge, the validity of Jaeger's conjecture in a broad inductively defined class of graphs. 1 Introduction One of the most challenging open problems in graph theory is the cycle double cover conjecture which was independently posed by Szekeres [14] and Seymour [13] in the seventies. It states that every bridgeless graph has a cycle double cover, that is, a system C of cycles such that each edge of the graph belongs to exactly two cycles of C. -
Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-Connected Graphs with Bounded Maximum Degrees
Georgia State University ScholarWorks @ Georgia State University Mathematics Dissertations Department of Mathematics and Statistics 5-11-2015 Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees Songling Shan Follow this and additional works at: https://scholarworks.gsu.edu/math_diss Recommended Citation Shan, Songling, "Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees." Dissertation, Georgia State University, 2015. https://scholarworks.gsu.edu/math_diss/23 This Dissertation is brought to you for free and open access by the Department of Mathematics and Statistics at ScholarWorks @ Georgia State University. It has been accepted for inclusion in Mathematics Dissertations by an authorized administrator of ScholarWorks @ Georgia State University. For more information, please contact [email protected]. Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees by Songling Shan Under the Direction of Guantao Chen, PhD ABSTRACT A tree T with no vertex of degree 2 is called a homeomorphically irreducible tree (HIT) and if T is spanning in a graph, then T is called a homeomorphically irreducible spanning tree (HIST). Albertson, Berman, Hutchinson and Thomassen asked if every triangulation of at least 4 vertices has a HIST and if every connected graph with each edge in at least two triangles contains a HIST. These two questions were restated as two conjectures by Archdeacon in 2009. The first part of this dissertation gives a proof for each of the two conjectures. The second part focuses on some problems about Halin graphs, which is a class of graphs closely related to HITs and HISTs. -
Configurations of Points and Lines
Configurations of Points and Lines "RANKO'RüNBAUM 'RADUATE3TUDIES IN-ATHEMATICS 6OLUME !MERICAN-ATHEMATICAL3OCIETY http://dx.doi.org/10.1090/gsm/103 Configurations of Points and Lines Configurations of Points and Lines Branko Grünbaum Graduate Studies in Mathematics Volume 103 American Mathematical Society Providence, Rhode Island Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 01A55, 01A60, 05–03, 05B30, 05C62, 51–03, 51A20, 51A45, 51E30, 52C30. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-103 Library of Congress Cataloging-in-Publication Data Gr¨unbaum, Branko. Configurations of points and lines / Branko Gr¨unbaum. p. cm. — (Graduate studies in mathematics ; v. 103) Includes bibliographical references and index. ISBN 978-0-8218-4308-6 (alk. paper) 1. Configurations. I. Title. QA607.G875 2009 516.15—dc22 2009000303 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society. -
Levi Graphs and Concurrence Graphs As Tools to Evaluate Designs
Levi graphs and concurrence graphs as tools to evaluate designs R. A. Bailey [email protected] The Norman Biggs Lecture, May 2012 1/35 What makes a block design good for experiments? I have v treatments that I want to compare. I have b blocks. Each block has space for k treatments (not necessarily distinct). How should I choose a block design? 2/35 Two designs with v = 5, b = 7, k = 3: which is better? Conventions: columns are blocks; order of treatments within each block is irrelevant; order of blocks is irrelevant. 1 1 1 1 2 2 2 1 1 1 1 2 2 2 2 3 3 4 3 3 4 1 3 3 4 3 3 4 3 4 5 5 4 5 5 2 4 5 5 4 5 5 binary non-binary A design is binary if no treatment occurs more than once in any block. 3/35 Two designs with v = 15, b = 7, k = 3: which is better? 1 1 2 3 4 5 6 1 1 1 1 1 1 1 2 4 5 6 10 11 12 2 4 6 8 10 12 14 3 7 8 9 13 14 15 3 5 7 9 11 13 15 replications differ by ≤ 1 queen-bee design The replication of a treatment is its number of occurrences. A design is a queen-bee design if there is a treatment that occurs in every block. 4/35 Two designs with v = 7, b = 7, k = 3: which is better? 1 2 3 4 5 6 7 1 2 3 4 5 6 7 2 3 4 5 6 7 1 2 3 4 5 6 7 1 4 5 6 7 1 2 3 3 4 5 6 7 1 2 balanced (2-design) non-balanced A binary design is balanced if every pair of distinct treaments occurs together in the same number of blocks. -
Graph Theory with Applications
GRAPH THEORY WITH APPLICATIONS J. A. Bondy and U. S. R. Murty Department of Combina tories and Optimization, University of Waterloo, Ontario, Canada NORfH-HOLLAND New York • Amsterdam • Oxford @J.A. Bondy and U.S.R. Muny 1976 First published in Great Britain 1976 by The Macmillan Press Ltd. First published in the U.S.A. 1976 by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, N.Y. 10017 Fifth Printing, 1982. Sole Distributor in the U.S.A: Elsevier Science Publishing Co., Inc. Library of Congress Cataloging in Publication Data Bondy, John Adrian. Graph theory with applications. Bibliography: p. lncludes index. 1. Graph theory. 1. Murty, U.S.R., joint author. II. Title. QA166.B67 1979 511 '.5 75-29826 ISBN 0.:444-19451-7 AU rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. Printed in the United States of America To our parents Preface This book is intended as an introduction to graph theory. Our aim bas been to present what we consider to be the basic material, together with a wide variety of applications, both to other branches of mathematics and to real-world problems. Included are simple new proofs of theorems of Brooks, Chvâtal, Tutte and Vizing. The applications have been carefully selected, and are treated in some depth. We have chosen to omit ail so-called 'applications' that employ just the language of graphs and no theory. The applications appearing at the end of each chapter actually make use of theory developed earlier in the same chapter. -
Math 7410 Graph Theory
Math 7410 Graph Theory Bogdan Oporowski Department of Mathematics Louisiana State University April 14, 2021 Definition of a graph Definition 1.1 A graph G is a triple (V,E, I) where ◮ V (or V (G)) is a finite set whose elements are called vertices; ◮ E (or E(G)) is a finite set disjoint from V whose elements are called edges; and ◮ I, called the incidence relation, is a subset of V E in which each edge is × in relation with exactly one or two vertices. v2 e1 v1 Example 1.2 ◮ V = v1, v2, v3, v4 e5 { } e2 e4 ◮ E = e1,e2,e3,e4,e5,e6,e7 { } e6 ◮ I = (v1,e1), (v1,e4), (v1,e5), (v1,e6), { v4 (v2,e1), (v2,e2), (v3,e2), (v3,e3), (v3,e5), e7 v3 e3 (v3,e6), (v4,e3), (v4,e4), (v4,e7) } Simple graphs Definition 1.3 ◮ Edges incident with just one vertex are loops. ◮ Edges incident with the same pair of vertices are parallel. ◮ Graphs with no parallel edges and no loops are called simple. v2 e1 v1 e5 e2 e4 e6 v4 e7 v3 e3 Edges of a simple graph can be described as v e1 v 2 1 two-element subsets of the vertex set. Example 1.4 e5 e2 e4 E = v1, v2 , v2, v3 , v3, v4 , e6 {{ } { } { } v1, v4 , v1, v3 . v4 { } { }} v e7 3 e3 Note 1.5 Graph Terminology Definition 1.6 ◮ The graph G is empty if V = , and is trivial if E = . ∅ ∅ ◮ The cardinality of the vertex-set of a graph G is called the order of G and denoted G . -
Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur Si Muove: Biomimetic Embedding of N-Tuple Helices in Spherical Polyhedra - /
Alternative view of segmented documents via Kairos 23 October 2017 | Draft Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur si muove: Biomimetic embedding of N-tuple helices in spherical polyhedra - / - Introduction Symbolic stars vs Strategic pillars; Polyhedra vs Helices; Logic vs Comprehension? Dynamic bonding patterns in n-tuple helices engendering n-fold rotating symbols Embedding the triple helix in a spherical octahedron Embedding the quadruple helix in a spherical cube Embedding the quintuple helix in a spherical dodecahedron and a Pentagramma Mirificum Embedding six-fold, eight-fold and ten-fold helices in appropriately encircled polyhedra Embedding twelve-fold, eleven-fold, nine-fold and seven-fold helices in appropriately encircled polyhedra Neglected recognition of logical patterns -- especially of opposition Dynamic relationship between polyhedra engendered by circles -- variously implying forms of unity Symbol rotation as dynamic essential to engaging with value-inversion References Introduction The contrast to the geocentric model of the solar system was framed by the Italian mathematician, physicist and philosopher Galileo Galilei (1564-1642). His much-cited phrase, " And yet it moves" (E pur si muove or Eppur si muove) was allegedly pronounced in 1633 when he was forced to recant his claims that the Earth moves around the immovable Sun rather than the converse -- known as the Galileo affair. Such a shift in perspective might usefully inspire the recognition that the stasis attributed so widely to logos and other much-valued cultural and heraldic symbols obscures the manner in which they imply a fundamental cognitive dynamic. Cultural symbols fundamental to the identity of a group might then be understood as variously moving and transforming in ways which currently elude comprehension. -
Binomial Partial Steiner Triple Systems Containing Complete Graphs
Graphs and Combinatorics DOI 10.1007/s00373-016-1681-3 ORIGINAL PAPER Binomial partial Steiner triple systems containing complete graphs Małgorzata Pra˙zmowska1 · Krzysztof Pra˙zmowski2 Received: 1 December 2014 / Revised: 25 January 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We propose a new approach to studies on partial Steiner triple systems con- sisting in determining complete graphs contained in them. We establish the structure which complete graphs yield in a minimal PSTS that contains them. As a by-product we introduce the notion of a binomial PSTS as a configuration with parameters of a minimal PSTS with a complete subgraph. A representation of binomial PSTS with at least a given number of its maximal complete subgraphs is given in terms of systems of perspectives. Finally, we prove that for each admissible integer there is a binomial PSTS with this number of maximal complete subgraphs. Keywords Binomial configuration · Generalized Desargues configuration · Complete graph Mathematics Subject Classification 05B30 · 05C51 · 05B40 1 Introduction In the paper we investigate the structure which (may) yield complete graphs contained in a (partial) Steiner triple system (in short: in a PSTS). Our problem is, in fact, a particular instance of a general question, investigated in the literature, which STS’s B Małgorzata Pra˙zmowska [email protected] Krzysztof Pra˙zmowski [email protected] 1 Faculty of Mathematics and Informatics, University of Białystok, ul. Ciołkowskiego 1M, 15-245 Białystok, Poland 2 Institute of Mathematics, University of Białystok, ul. Ciołkowskiego 1M, 15-245 Białystok, Poland 123 Graphs and Combinatorics (more generally: which PSTS’s) contain/do not contain a configuration of a prescribed type. -
Some Topics Concerning Graphs, Signed Graphs and Matroids
SOME TOPICS CONCERNING GRAPHS, SIGNED GRAPHS AND MATROIDS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Vaidyanathan Sivaraman, M.S. Graduate Program in Mathematics The Ohio State University 2012 Dissertation Committee: Prof. Neil Robertson, Advisor Prof. Akos´ Seress Prof. Matthew Kahle ABSTRACT We discuss well-quasi-ordering in graphs and signed graphs, giving two short proofs of the bounded case of S. B. Rao's conjecture. We give a characterization of graphs whose bicircular matroids are signed-graphic, thus generalizing a theorem of Matthews from the 1970s. We prove a recent conjecture of Zaslavsky on the equality of frus- tration number and frustration index in a certain class of signed graphs. We prove that there are exactly seven signed Heawood graphs, up to switching isomorphism. We present a computational approach to an interesting conjecture of D. J. A. Welsh on the number of bases of matroids. We then move on to study the frame matroids of signed graphs, giving explicit signed-graphic representations of certain families of matroids. We also discuss the cycle, bicircular and even-cycle matroid of a graph and characterize matroids arising as two different such structures. We study graphs in which any two vertices have the same number of common neighbors, giving a quick proof of Shrikhande's theorem. We provide a solution to a problem of E. W. Dijkstra. Also, we discuss the flexibility of graphs on the projective plane. We conclude by men- tioning partial progress towards characterizing signed graphs whose frame matroids are transversal, and some miscellaneous results. -
COMBINATORICS, Volume
http://dx.doi.org/10.1090/pspum/019 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume XIX COMBINATORICS AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 1971 Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society Held at the University of California Los Angeles, California March 21-22, 1968 Prepared by the American Mathematical Society under National Science Foundation Grant GP-8436 Edited by Theodore S. Motzkin AMS 1970 Subject Classifications Primary 05Axx, 05Bxx, 05Cxx, 10-XX, 15-XX, 50-XX Secondary 04A20, 05A05, 05A17, 05A20, 05B05, 05B15, 05B20, 05B25, 05B30, 05C15, 05C99, 06A05, 10A45, 10C05, 14-XX, 20Bxx, 20Fxx, 50A20, 55C05, 55J05, 94A20 International Standard Book Number 0-8218-1419-2 Library of Congress Catalog Number 74-153879 Copyright © 1971 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government May not be produced in any form without permission of the publishers Leo Moser (1921-1970) was active and productive in various aspects of combin• atorics and of its applications to number theory. He was in close contact with those with whom he had common interests: we will remember his sparkling wit, the universality of his anecdotes, and his stimulating presence. This volume, much of whose content he had enjoyed and appreciated, and which contains the re• construction of a contribution by him, is dedicated to his memory. CONTENTS Preface vii Modular Forms on Noncongruence Subgroups BY A. O. L. ATKIN AND H. P. F. SWINNERTON-DYER 1 Selfconjugate Tetrahedra with Respect to the Hermitian Variety xl+xl + *l + ;cg = 0 in PG(3, 22) and a Representation of PG(3, 3) BY R. -
Zadání Bakalářské Práce
ČESKÉ VYSOKÉ UČENÍ TECHNICKÉ V PRAZE FAKULTA INFORMAČNÍCH TECHNOLOGIÍ ZADÁNÍ BAKALÁŘSKÉ PRÁCE Název: Rekurzivně konstruovatelné grafy Student: Anežka Štěpánková Vedoucí: RNDr. Jiřina Scholtzová, Ph.D. Studijní program: Informatika Studijní obor: Teoretická informatika Katedra: Katedra teoretické informatiky Platnost zadání: Do konce letního semestru 2016/17 Pokyny pro vypracování Souvislé rekurentně zadané grafy jsou souvislé grafy, pro které existuje postup, jak ze zadaného grafu odvodit graf nový. V předmětu BI-GRA jste se s jednoduchými rekurentními grafy setkali, vždy šly popsat lineárními rekurentními rovnicemi s konstantními koeficienty. Existuje klasifikace, která definuje třídu rekurzivně konstruovatelných grafů [1]. Podobně jako u grafů v BI-GRA zjistěte, zda lze některé z kategorií rekurzivně konstruovatelných grafů popsat rekurentními rovnicemi. 1. Seznamte se s rekurentně zadanými grafy. 2. Seznamte se s rekurzivně konstruovatelnými grafy a nastudujte články o nich. 3. Vyberte si některé z kategorií rekurzivně konstruovatelných grafů [1]. 4. Pro vybrané kategorie najděte způsob, jak je popsat využitím rekurentních rovnic. Seznam odborné literatury [1] Gross, Jonathan L., Jay Yellen, Ping Zhang. Handbook of Graph Theory, 2nd ed, Boca Raton: CRC, 2013. ISBN: 978-1-4398-8018-0 doc. Ing. Jan Janoušek, Ph.D. prof. Ing. Pavel Tvrdík, CSc. vedoucí katedry děkan V Praze dne 18. února 2016 České vysoké učení technické v Praze Fakulta informačních technologií Katedra Teoretické informatiky Bakalářská práce Rekurzivně konstruovatelné grafy Anežka Štěpánková Vedoucí práce: RNDr. Jiřina Scholtzová Ph.D. 12. května 2017 Poděkování Děkuji vedoucí práce RNDr. Jiřině Scholtzové Ph.D., která mi ochotně vyšla vstříc s volbou tématu a v průběhu tvorby práce vždy dobře poradila a na- směrovala mne správným směrem. -
Instituto De Computação ✄✂ ✁✂✄✁✂✄✁✂ ✄✁ ✄✂✁ ✄✂ ✁✄✂ ✁ ✂✁✂✁✂✁✂✁✂✁✂✁✂✁ Universidade Estadual De Campinas
✄✄ ✄ ✄ ✄ ✄✄ ✄✂ ✁✂✄✁✂ ✄✁✂ ✄✁ ✄✂ ✁✄ ✂✄✁✂ ✁ ✄✂ ✄✁ ✄✂ ✄✁✂ ✁✂✄✁✂ ✁✄✂ ✁✄✂ ✁ INSTITUTO DE COMPUTAÇÃO ✄✂ ✁✂✄✁✂✄ ✁✂✄ ✁✄ ✂ ✁✄✂ ✁✄✂ ✁ ✂✁✂✁✂✁✂✁✂✁✂✁✂✁ UNIVERSIDADE ESTADUAL DE CAMPINAS Colorings and crossings Atílio G. Luiz R. Bruce Richter Technical Report - IC-13-18 - Relatório Técnico August - 2013 - Agosto The contents of this report are the sole responsibility of the authors. O conteúdo do presente relatório é de única responsabilidade dos autores. Colorings and crossings∗ At´ılioG. Luiz† R. Bruce Richter‡ Abstract In 2007, Albertson conjectured that if a graph G has chromatic number k, then the crossing number of G is at least the crossing number of the complete graph with k vertices. To date, two papers were written on this subject trying to solve the conjecture for an arbitrary k-chromatic graph G, and after much effort the conjecture was proved true for k ≤ 16. In this report, we present an overview of topics related to Albertson’s Conjecture, such as: the crossing number problem, lower bounds on crossing number, lower bounds on the number of edges of color-critical graphs, and coloring of graphs with small crossing number and small clique number. While investigating Albertson’s conjecture, J. Bar´at and G. T´oth proposed the following conjecture involving color- critical graphs: for every positive integer c, there exists a bound k(c) such that for any k, where k ≥ k(c), any k-critical graph on k + c vertices has a subdivision of Kk. In Section 9, we present counterexamples to this conjecture for every c ≥ 6 and we prove that the conjecture is valid for c = 5. 1 Introduction A graph G is planar if it can be drawn in the plane so that its edges intersect only at their ends.