There Are No Cubic Graphs on 26 Vertices with Crossing Number 10 Or 11 3
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
Non-Hamiltonian 3–Regular Graphs with Arbitrary Girth
Universal Journal of Applied Mathematics 2(1): 72-78, 2014 DOI: 10.13189/ujam.2014.020111 http://www.hrpub.org Non-Hamiltonian 3{Regular Graphs with Arbitrary Girth M. Haythorpe School of Computer Science, Engineering and Mathematics, Flinders University, Australia ∗Corresponding Author: michael.haythorpe@flinders.edu.au Copyright ⃝c 2014 Horizon Research Publishing All rights reserved. Abstract It is well known that 3{regular graphs with arbitrarily large girth exist. Three constructions are given that use the former to produce non-Hamiltonian 3{regular graphs without reducing the girth, thereby proving that such graphs with arbitrarily large girth also exist. The resulting graphs can be 1{, 2{ or 3{edge-connected de- pending on the construction chosen. From the constructions arise (naive) upper bounds on the size of the smallest non-Hamiltonian 3{regular graphs with particular girth. Several examples are given of the smallest such graphs for various choices of girth and connectedness. Keywords Girth, Cages, Cubic, 3-Regular, Prisons, Hamiltonian, non-Hamiltonian 1 Introduction Consider a k{regular graph Γ with girth g, containing N vertices. Then Γ is said to be a (k; g){cage if and only if all other k{regular graphs with girth g contain N or more vertices. The study of cages, or cage graphs, goes back to Tutte [15], who later gave lower bounds on n(k; g), the number of vertices in a (k; g){cage [16]. The best known upper bounds on n(k; g) were obtained by Sauer [14] around the same time. Since that time, the advent of vast computational power has enabled large-scale searches such as those conducted by McKay et al [10], and Royle [13] who maintains a webpage with examples of known cages. -
Maximizing the Order of a Regular Graph of Given Valency and Second Eigenvalue∗
SIAM J. DISCRETE MATH. c 2016 Society for Industrial and Applied Mathematics Vol. 30, No. 3, pp. 1509–1525 MAXIMIZING THE ORDER OF A REGULAR GRAPH OF GIVEN VALENCY AND SECOND EIGENVALUE∗ SEBASTIAN M. CIOABA˘ †,JACKH.KOOLEN‡, HIROSHI NOZAKI§, AND JASON R. VERMETTE¶ Abstract. From Alon√ and Boppana, and Serre, we know that for any given integer k ≥ 3 and real number λ<2 k − 1, there are only finitely many k-regular graphs whose second largest eigenvalue is at most λ. In this paper, we investigate the largest number of vertices of such graphs. Key words. second eigenvalue, regular graph, expander AMS subject classifications. 05C50, 05E99, 68R10, 90C05, 90C35 DOI. 10.1137/15M1030935 1. Introduction. For a k-regular graph G on n vertices, we denote by λ1(G)= k>λ2(G) ≥ ··· ≥ λn(G)=λmin(G) the eigenvalues of the adjacency matrix of G. For a general reference on the eigenvalues of graphs, see [8, 17]. The second eigenvalue of a regular graph is a parameter of interest in the study of graph connectivity and expanders (see [1, 8, 23], for example). In this paper, we investigate the maximum order v(k, λ) of a connected k-regular graph whose second largest eigenvalue is at most some given parameter λ. As a consequence of work of Alon and Boppana and of Serre√ [1, 11, 15, 23, 24, 27, 30, 34, 35, 40], we know that v(k, λ) is finite for λ<2 k − 1. The recent result of Marcus, Spielman, and Srivastava [28] showing the existence of infinite families of√ Ramanujan graphs of any degree at least 3 implies that v(k, λ) is infinite for λ ≥ 2 k − 1. -
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. -
The Intriguing Human Preference for a Ternary Patterned Reality
75 Kragujevac J. Sci. 27 (2005) 75-114. THE INTRIGUING HUMAN PREFERENCE FOR A TERNARY PATTERNED REALITY Lionello Pogliani,* Douglas J. Klein,‡ and Alexandru T. Balaban¥ *Dipartimento di Chimica Università della Calabria, 87030 Rende (CS), Italy; ‡Texas A&M University at Galveston, Galveston, Texas, 77553-1675, USA; ¥ Texas A&M University at Galveston, Galveston, Texas, 77551, USA E-mails: [email protected]; [email protected]; balabana @tamug.tamu.edu (Received February 10, 2005) Three paths through the high spring grass / One is the quicker / and I take it (Haikku by Yosa Buson, Spring wind on the river bank Kema) How did number theory degrade to numerology? How did numerology influence different aspects of human creation? How could some numbers assume such an extraordinary meaning? Why some numbers appear so often throughout the fabric of reality? Has reality really a preference for a ternary pattern? Present excursion through the ‘life’ and ‘deeds’ of this pattern and of its parent number in different fields of human culture try to offer an answer to some of these questions giving at the same time a glance of the strange preference of the human mind for a patterned reality based on number three, which, like the 'unary' and 'dualistic' patterned reality, but in a more emphatic way, ended up assuming an archetypical character in the intellectual sphere of humanity. Introduction Three is not only the sole number that is the sum of all the natural numbers less than itself, but it lies between arguably the two most important irrational and transcendental numbers of all mathematics, i.e., e < 3 < π. -
On the Cycle Double Cover Problem
On The Cycle Double Cover Problem Ali Ghassâb1 Dedicated to Prof. E.S. Mahmoodian Abstract In this paper, for each graph , a free edge set is defined. To study the existence of cycle double cover, the naïve cycle double cover of have been defined and studied. In the main theorem, the paper, based on the Kuratowski minor properties, presents a condition to guarantee the existence of a naïve cycle double cover for couple . As a result, the cycle double cover conjecture has been concluded. Moreover, Goddyn’s conjecture - asserting if is a cycle in bridgeless graph , there is a cycle double cover of containing - will have been proved. 1 Ph.D. student at Sharif University of Technology e-mail: [email protected] Faculty of Math, Sharif University of Technology, Tehran, Iran 1 Cycle Double Cover: History, Trends, Advantages A cycle double cover of a graph is a collection of its cycles covering each edge of the graph exactly twice. G. Szekeres in 1973 and, independently, P. Seymour in 1979 conjectured: Conjecture (cycle double cover). Every bridgeless graph has a cycle double cover. Yielded next data are just a glimpse review of the history, trend, and advantages of the research. There are three extremely helpful references: F. Jaeger’s survey article as the oldest one, and M. Chan’s survey article as the newest one. Moreover, C.Q. Zhang’s book as a complete reference illustrating the relative problems and rather new researches on the conjecture. A number of attacks, to prove the conjecture, have been happened. Some of them have built new approaches and trends to study. -
The History of Degenerate (Bipartite) Extremal Graph Problems
The history of degenerate (bipartite) extremal graph problems Zolt´an F¨uredi and Mikl´os Simonovits May 15, 2013 Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary [email protected] and [email protected] Abstract This paper is a survey on Extremal Graph Theory, primarily fo- cusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions. 1 Contents 1 Introduction 4 1.1 Some central theorems of the field . 5 1.2 Thestructureofthispaper . 6 1.3 Extremalproblems ........................ 8 1.4 Other types of extremal graph problems . 10 1.5 Historicalremarks . .. .. .. .. .. .. 11 arXiv:1306.5167v2 [math.CO] 29 Jun 2013 2 The general theory, classification 12 2.1 The importance of the Degenerate Case . 14 2.2 The asymmetric case of Excluded Bipartite graphs . 15 2.3 Reductions:Hostgraphs. 16 2.4 Excluding complete bipartite graphs . 17 2.5 Probabilistic lower bound . 18 1 Research supported in part by the Hungarian National Science Foundation OTKA 104343, and by the European Research Council Advanced Investigators Grant 267195 (ZF) and by the Hungarian National Science Foundation OTKA 101536, and by the European Research Council Advanced Investigators Grant 321104. (MS). 1 F¨uredi-Simonovits: Degenerate (bipartite) extremal graph problems 2 2.6 Classification of extremal problems . 21 2.7 General conjectures on bipartite graphs . 23 3 Excluding complete bipartite graphs 24 3.1 Bipartite C4-free graphs and the Zarankiewicz problem . 24 3.2 Finite Geometries and the C4-freegraphs . -
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. -
Octonion Multiplication and Heawood's
CONFLUENTES MATHEMATICI Bruno SÉVENNEC Octonion multiplication and Heawood’s map Tome 5, no 2 (2013), p. 71-76. <http://cml.cedram.org/item?id=CML_2013__5_2_71_0> © Les auteurs et Confluentes Mathematici, 2013. Tous droits réservés. L’accès aux articles de la revue « Confluentes Mathematici » (http://cml.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://cml.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation á fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Confluentes Math. 5, 2 (2013) 71-76 OCTONION MULTIPLICATION AND HEAWOOD’S MAP BRUNO SÉVENNEC Abstract. In this note, the octonion multiplication table is recovered from a regular tesse- lation of the equilateral two timensional torus by seven hexagons, also known as Heawood’s map. Almost any article or book dealing with Cayley-Graves algebra O of octonions (to be recalled shortly) has a picture like the following Figure 0.1 representing the so-called ‘Fano plane’, which will be denoted by Π, together with some cyclic ordering on each of its ‘lines’. The Fano plane is a set of seven points, in which seven three-point subsets called ‘lines’ are specified, such that any two points are contained in a unique line, and any two lines intersect in a unique point, giving a so-called (combinatorial) projective plane [8,7]. -
Large Girth Graphs with Bounded Diameter-By-Girth Ratio 3
LARGE GIRTH GRAPHS WITH BOUNDED DIAMETER-BY-GIRTH RATIO GOULNARA ARZHANTSEVA AND ARINDAM BISWAS Abstract. We provide an explicit construction of finite 4-regular graphs (Γk)k∈N with girth Γk k → ∞ as k and diam Γ 6 D for some D > 0 and all k N. For each dimension n > 2, we find a → ∞ girth Γk ∈ pair of matrices in SLn(Z) such that (i) they generate a free subgroup, (ii) their reductions mod p generate SLn(Fp) for all sufficiently large primes p, (iii) the corresponding Cayley graphs of SLn(Fp) have girth at least cn log p for some cn > 0. Relying on growth results (with no use of expansion properties of the involved graphs), we observe that the diameter of those Cayley graphs is at most O(log p). This gives infinite sequences of finite 4-regular Cayley graphs as in the title. These are the first explicit examples in all dimensions n > 2 (all prior examples were in n = 2). Moreover, they happen to be expanders. Together with Margulis’ and Lubotzky-Phillips-Sarnak’s classical con- structions, these new graphs are the only known explicit large girth Cayley graph expanders with bounded diameter-by-girth ratio. 1. Introduction The girth of a graph is the edge-length of its shortest non-trivial cycle (it is assigned to be infinity for an acyclic graph). The diameter of a graph is the greatest edge-length distance between any pair of its vertices. We regard a graph Γ as a sequence of its connected components Γ = (Γk)k∈N each of which is endowed with the edge-length distance. -
On the Levi Graph of Point-Line Configurations 895
inv lve a journal of mathematics On the Levi graph of point-line configurations Jessica Hauschild, Jazmin Ortiz and Oscar Vega msp 2015 vol. 8, no. 5 INVOLVE 8:5 (2015) msp dx.doi.org/10.2140/involve.2015.8.893 On the Levi graph of point-line configurations Jessica Hauschild, Jazmin Ortiz and Oscar Vega (Communicated by Joseph A. Gallian) We prove that the well-covered dimension of the Levi graph of a point-line configuration with v points, b lines, r lines incident with each point, and every line containing k points is equal to 0, whenever r > 2. 1. Introduction The concept of the well-covered space of a graph was first introduced by Caro, Ellingham, Ramey, and Yuster[Caro et al. 1998; Caro and Yuster 1999] as an effort to generalize the study of well-covered graphs. Brown and Nowakowski [2005] continued the study of this object and, among other things, provided several examples of graphs featuring odd behaviors regarding their well-covered spaces. One of these special situations occurs when the well-covered space of the graph is trivial, i.e., when the graph is anti-well-covered. In this work, we prove that almost all Levi graphs of configurations in the family of the so-called .vr ; bk/- configurations (see Definition 3) are anti-well-covered. We start our exposition by providing the following definitions and previously known results. Any introductory concepts we do not present here may be found in the books by Bondy and Murty[1976] and Grünbaum[2009]. We consider only simple and undirected graphs.