Combinatorial Design Theory - Notes
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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. -
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 . -
Combinatorial Design University of Southern California Non-Parametric Computational Design Strategies
Jose Sanchez Combinatorial design University of Southern California Non-parametric computational design strategies 1 ABSTRACT This paper outlines a framework and conceptualization of combinatorial design. Combinatorial 1 Design – Year 3 – Pattern of one design is a term coined to describe non-parametric design strategies that focus on the permutation, unit in two scales. Developed by combinatorial design within a game combination and patterning of discrete units. These design strategies differ substantially from para- engine. metric design strategies as they do not operate under continuous numerical evaluations, intervals or ratios but rather finite discrete sets. The conceptualization of this term and the differences with other design strategies are portrayed by the work done in the last 3 years of research at University of Southern California under the Polyomino agenda. The work, conducted together with students, has studied the use of discrete sets and combinatorial strategies within virtual reality environments to allow for an enhanced decision making process, one in which human intuition is coupled to algo- rithmic intelligence. The work of the research unit has been sponsored and tested by the company Stratays for ongoing research on crowd-sourced design. 44 INTRODUCTION—OUTSIDE THE PARAMETRIC UMBRELLA To start, it is important that we understand that the use of the terms parametric and combinatorial that will be used in this paper will come from an architectural and design background, as the association of these terms in mathematics and statistics might have a different connotation. There are certainly lessons and a direct relation between the term ‘combinatorial’ as used in this paper and the field of combinatorics and permutations in mathematics. -
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. -
The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 Email: [email protected]
The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 email: [email protected] In this column we review the following books. 1. Combinatorial Designs: Constructions and Analysis by Douglas R. Stinson. Review by Gregory Taylor. A combinatorial design is a set of sets of (say) {1, . , n} that have various properties, such as that no two of them have a large intersection. For what parameters do such designs exist? This is an interesting question that touches on many branches of math for both origin and application. 2. Combinatorics of Permutations by Mikl´osB´ona. Review by Gregory Taylor. Usually permutations are viewed as a tool in combinatorics. In this book they are considered as objects worthy of study in and of themselves. 3. Enumerative Combinatorics by Charalambos A. Charalambides. Review by Sergey Ki- taev, 2008. Enumerative combinatorics is a branch of combinatorics concerned with counting objects satisfying certain criteria. This is a far reaching and deep question. 4. Geometric Algebra for Computer Science by L. Dorst, D. Fontijne, and S. Mann. Review by B. Fasy and D. Millman. How can we view Geometry in terms that a computer can understand and deal with? This book helps answer that question. 5. Privacy on the Line: The Politics of Wiretapping and Encryption by Whitfield Diffie and Susan Landau. Review by Richard Jankowski. What is the status of your privacy given current technology and law? Read this book and find out! Books I want Reviewed If you want a FREE copy of one of these books in exchange for a review, then email me at gasarchcs.umd.edu Reviews need to be in LaTeX, LaTeX2e, or Plaintext. -
Geometry, Combinatorial Designs and Cryptology Fourth Pythagorean Conference
Geometry, Combinatorial Designs and Cryptology Fourth Pythagorean Conference Sunday 30 May to Friday 4 June 2010 Index of Talks and Abstracts Main talks 1. Simeon Ball, On subsets of a finite vector space in which every subset of basis size is a basis 2. Simon Blackburn, Honeycomb arrays 3. G`abor Korchm`aros, Curves over finite fields, an approach from finite geometry 4. Cheryl Praeger, Basic pregeometries 5. Bernhard Schmidt, Finiteness of circulant weighing matrices of fixed weight 6. Douglas Stinson, Multicollision attacks on iterated hash functions Short talks 1. Mari´en Abreu, Adjacency matrices of polarity graphs and of other C4–free graphs of large size 2. Marco Buratti, Combinatorial designs via factorization of a group into subsets 3. Mike Burmester, Lightweight cryptographic mechanisms based on pseudorandom number generators 4. Philippe Cara, Loops, neardomains, nearfields and sets of permutations 5. Ilaria Cardinali, On the structure of Weyl modules for the symplectic group 6. Bill Cherowitzo, Parallelisms of quadrics 7. Jan De Beule, Large maximal partial ovoids of Q−(5, q) 8. Bart De Bruyn, On extensions of hyperplanes of dual polar spaces 1 9. Frank De Clerck, Intriguing sets of partial quadrangles 10. Alice Devillers, Symmetry properties of subdivision graphs 11. Dalibor Froncek, Decompositions of complete bipartite graphs into generalized prisms 12. Stelios Georgiou, Self-dual codes from circulant matrices 13. Robert Gilman, Cryptology of infinite groups 14. Otokar Groˇsek, The number of associative triples in a quasigroup 15. Christoph Hering, Latin squares, homologies and Euler’s conjecture 16. Leanne Holder, Bilinear star flocks of arbitrary cones 17. Robert Jajcay, On the geometry of cages 18. -
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. -
Block Designs and Graph Theory*
JOURNAL OF COMBINATORIAL qtIliORY 1, 132-148 (1966) Block Designs and Graph Theory* JANE W. DI PAOLA The City University of New York Comnumicated by R.C. Bose INTRODUCTION The purpose of this paper is to demonstrate the relation of balanced incomplete block designs to certain concepts of graph theory. The set of blocks of a balanced incomplete block design with Z -- 1 is shown to be related to a maximum internally stable set of vertices of a suitably defined graph. The development yields also an upper bound for the internal stability number of a large subclass of a class of graphs which we call "graphs on binomial coefficients." In a different but related context every balanced incomplete block design with 2 -- 1 is shown to be a solution of a suitably defined irreflexive relation. Some examples of relativizations and extensions of solutions of irreflexive relations (as developed by Richardson [13-15]) are generated as a result of the concepts derived. PRELIMINARY RESULTS A balanced incomplete block design (BIBD) is a set of v elements arranged in b blocks of k elements each in such a way that each element occurs r times and each unordered pair of distinct elements determines ,~ distinct blocks. The v, b, r, k, 2 are called the parameters of the design. * Research on which this paper is based supported by the U. S. Army Research Office-Durham under Contract No. DA-31-124-ARO-D-366. 132 BLOCK DESIGNS AND GRAPH THEORY 133 Following a suggestion implied by Berge [2] we introduce the DEHNmON. Agraph on the binomial coefficient (k)with edge pa- rameter 2, written G rr)k 4, is raphwhosevert ces ar t e (;).siDle k-tuples which can be formed from v elements and having as adjacent vertices those pairs of vertices which have more than 2 and less than k elements in common. -
Modules Over Semisymmetric Quasigroups
Modules over semisymmetric quasigroups Alex W. Nowak Abstract. The class of semisymmetric quasigroups is determined by the iden- tity (yx)y = x. We prove that the universal multiplication group of a semisym- metric quasigroup Q is free over its underlying set and then specify the point- stabilizers of an action of this free group on Q. A theorem of Smith indicates that Beck modules over semisymmetric quasigroups are equivalent to modules over a quotient of the integral group algebra of this stabilizer. Implementing our description of the quotient ring, we provide some examples of semisym- metric quasigroup extensions. Along the way, we provide an exposition of the quasigroup module theory in more general settings. 1. Introduction A quasigroup (Q, ·, /, \) is a set Q equipped with three binary operations: · denoting multiplication, / right division, and \ left division; these operations satisfy the identities (IL) y\(y · x)= x; (IR) x = (x · y)/y; (SL) y · (y\x)= x; (SR) x = (x/y) · y. Whenever possible, we convey multiplication of quasigroup elements by concatena- tion. As a class of algebras defined by identities, quasigroups constitute a variety (in the universal-algebraic sense), denoted by Q. Moreover, we have a category of quasigroups (also denoted Q), the morphisms of which are quasigroup homomor- phisms respecting the three operations. arXiv:1908.06364v1 [math.RA] 18 Aug 2019 If (G, ·) is a group, then defining x/y := x · y−1 and x\y := x−1 · y furnishes a quasigroup (G, ·, /, \). That is, all groups are quasigroups, and because of this fact, constructions in the representation theory of quasigroups often take their cue from familiar counterparts in group theory. -
7 Combinatorial Geometry
7 Combinatorial Geometry Planes Incidence Structure: An incidence structure is a triple (V; B; ∼) so that V; B are disjoint sets and ∼ is a relation on V × B. We call elements of V points, elements of B blocks or lines and we associate each line with the set of points incident with it. So, if p 2 V and b 2 B satisfy p ∼ b we say that p is contained in b and write p 2 b and if b; b0 2 B we let b \ b0 = fp 2 P : p ∼ b; p ∼ b0g. Levi Graph: If (V; B; ∼) is an incidence structure, the associated Levi Graph is the bipartite graph with bipartition (V; B) and incidence given by ∼. Parallel: We say that the lines b; b0 are parallel, and write bjjb0 if b \ b0 = ;. Affine Plane: An affine plane is an incidence structure P = (V; B; ∼) which satisfies the following properties: (i) Any two distinct points are contained in exactly one line. (ii) If p 2 V and ` 2 B satisfy p 62 `, there is a unique line containing p and parallel to `. (iii) There exist three points not all contained in a common line. n Line: Let ~u;~v 2 F with ~v 6= 0 and let L~u;~v = f~u + t~v : t 2 Fg. Any set of the form L~u;~v is called a line in Fn. AG(2; F): We define AG(2; F) to be the incidence structure (V; B; ∼) where V = F2, B is the set of lines in F2 and if v 2 V and ` 2 B we define v ∼ ` if v 2 `. -
ON the POLYHEDRAL GEOMETRY of T–DESIGNS
ON THE POLYHEDRAL GEOMETRY OF t{DESIGNS A thesis presented to the faculty of San Francisco State University In partial fulfilment of The Requirements for The Degree Master of Arts In Mathematics by Steven Collazos San Francisco, California August 2013 Copyright by Steven Collazos 2013 CERTIFICATION OF APPROVAL I certify that I have read ON THE POLYHEDRAL GEOMETRY OF t{DESIGNS by Steven Collazos and that in my opinion this work meets the criteria for approving a thesis submitted in partial fulfillment of the requirements for the degree: Master of Arts in Mathematics at San Francisco State University. Matthias Beck Associate Professor of Mathematics Felix Breuer Federico Ardila Associate Professor of Mathematics ON THE POLYHEDRAL GEOMETRY OF t{DESIGNS Steven Collazos San Francisco State University 2013 Lisonek (2007) proved that the number of isomorphism types of t−(v; k; λ) designs, for fixed t, v, and k, is quasi{polynomial in λ. We attempt to describe a region in connection with this result. Specifically, we attempt to find a region F of Rd with d the following property: For every x 2 R , we have that jF \ Gxj = 1, where Gx denotes the G{orbit of x under the action of G. As an application, we argue that our construction could help lead to a new combinatorial reciprocity theorem for the quasi{polynomial counting isomorphism types of t − (v; k; λ) designs. I certify that the Abstract is a correct representation of the content of this thesis. Chair, Thesis Committee Date ACKNOWLEDGMENTS I thank my advisors, Dr. Matthias Beck and Dr.