The History of Degenerate (Bipartite) Extremal Graph Problems
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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. -
Three Conjectures in Extremal Spectral Graph Theory
Three conjectures in extremal spectral graph theory Michael Tait and Josh Tobin June 6, 2016 Abstract We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of P2 and Pn−2 is the unique graph of maximum spectral radius over all planar graphs. This was conjectured by Boots and Royle in 1991 and independently by Cao and Vince in 1993. Similarly, we prove a conjecture of Cvetkovi´cand Rowlinson from 1990 stating that the unique outerplanar graph of maximum spectral radius is the join of a vertex and Pn−1. Finally, we prove a conjecture of Aouchiche et al from 2008 stating that a pineapple graph is the unique connected graph maximizing the spectral radius minus the average degree. To prove our theorems, we use the leading eigenvector of a purported extremal graph to deduce structural properties about that graph. Using this setup, we give short proofs of several old results: Mantel's Theorem, Stanley's edge bound and extensions, the K}ovari-S´os-Tur´anTheorem applied to ex (n; K2;t), and a partial solution to an old problem of Erd}oson making a triangle-free graph bipartite. 1 Introduction Questions in extremal graph theory ask to maximize or minimize a graph invariant over a fixed family of graphs. Perhaps the most well-studied problems in this area are Tur´an-type problems, which ask to maximize the number of edges in a graph which does not contain fixed forbidden subgraphs. Over a century old, a quintessential example of this kind of result is Mantel's theorem, which states that Kdn=2e;bn=2c is the unique graph maximizing the number of edges over all triangle-free graphs. -
Forbidding Subgraphs
Graph Theory and Additive Combinatorics Lecturer: Prof. Yufei Zhao 2 Forbidding subgraphs 2.1 Mantel’s theorem: forbidding a triangle We begin our discussion of extremal graph theory with the following basic question. Question 2.1. What is the maximum number of edges in an n-vertex graph that does not contain a triangle? Bipartite graphs are always triangle-free. A complete bipartite graph, where the vertex set is split equally into two parts (or differing by one vertex, in case n is odd), has n2/4 edges. Mantel’s theorem states that we cannot obtain a better bound: Theorem 2.2 (Mantel). Every triangle-free graph on n vertices has at W. Mantel, "Problem 28 (Solution by H. most bn2/4c edges. Gouwentak, W. Mantel, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff). Wiskundige Opgaven 10, 60 —61, 1907. We will give two proofs of Theorem 2.2. Proof 1. G = (V E) n m Let , a triangle-free graph with vertices and x edges. Observe that for distinct x, y 2 V such that xy 2 E, x and y N(x) must not share neighbors by triangle-freeness. Therefore, d(x) + d(y) ≤ n, which implies that d(x)2 = (d(x) + d(y)) ≤ mn. ∑ ∑ N(y) x2V xy2E y On the other hand, by the handshake lemma, ∑x2V d(x) = 2m. Now by the Cauchy–Schwarz inequality and the equation above, Adjacent vertices have disjoint neigh- borhoods in a triangle-free graph. !2 ! 4m2 = ∑ d(x) ≤ n ∑ d(x)2 ≤ mn2; x2V x2V hence m ≤ n2/4. -
Extremal Graph Theory
Extremal Graph Theory Ajit A. Diwan Department of Computer Science and Engineering, I. I. T. Bombay. Email: [email protected] Basic Question • Let H be a fixed graph. • What is the maximum number of edges in a graph G with n vertices that does not contain H as a subgraph? • This number is denoted ex(n,H). • A graph G with n vertices and ex(n,H) edges that does not contain H is called an extremal graph for H. Mantel’s Theorem (1907) n2 ex(n, K ) 3 4 • The only extremal graph for a triangle is the complete bipartite graph with parts of nearly equal sizes. Complete Bipartite graph Turan’s theorem (1941) t 2 ex(n, K ) n2 t 2(t 1) • Equality holds when n is a multiple of t-1. • The only extremal graph is the complete (t-1)- partite graph with parts of nearly equal sizes. Complete Multipartite Graph Proofs of Turan’s theorem • Many different proofs. • Use different techniques. • Techniques useful in proving other results. • Algorithmic applications. • “BOOK” proofs. Induction • The result is trivial if n <= t-1. • Suppose n >= t and consider a graph G with maximum number of edges and no Kt. • G must contain a Kt-1. • Delete all vertices in Kt-1. • The remaining graph contains at most t 2 (n t 1)2 edges. 2(t 1) Induction • No vertex outside Kt-1 can be joined to all vertices of Kt-1. • Total number of edges is at most t 2 (t 1)(t 2) (n t 1)2 2(t 1) 2 t 2 (n t 1)(t 2) n2 2(t 1) Greedy algorithm • Consider any extremal graph and let v be a vertex with maximum degree ∆. -
An Extremal Problem for Complete Bipartite Graphs
Studia Seientiarum Mathematicarum Hungarica 23 (1988), 319-326 AN EXTREMAL PROBLEM FOR COMPLETE BIPARTITE GRAPHS P. ERDŐS, R. J. FAUDREE, C . C . ROUSSEAU and R . H. SCHELP Dedicated to the memory of Paul Turán Abstract Define f(n, k) to be the largest integer q such that for every graph G of order n and size q, G contains every complete bipartite graph K u, ,, with a+h=n-k . We obtain (i) exact values for f(n, 0) and f(n, 1), (ii) upper and lower bounds for f(n, k) when ku2 is fixed and n is large, and (iii) an upper bound for f(n, lenl) . 1 . Introduction Extremal graph theory, which was initiated by Turán in 1941 [4], is still the source of many interesting and difficult problems . The standard problem is to deter- mine f(n, G), the smallest integer q such that every graph with n vertices and q edges contains a subgraph isomorphic to G . It is striking that whereas Turán completely determined f(n, Km), there is much which is as yet unknown concerning f(n, Ka, b). In this paper, we consider a variant of the extremal problem for complete bipartite graphs . In this variant we ask how many edges must be deleted from K„ so that the resulting graph no longer contains Ka,, for some pair (a, b) with a+b=in . Specifically, we seek to determine an extremal function f(n, k) defined as follows . For in ::- 1, let B,,, denote the class of all graphs G such that GDKa , b for every pair (a, b) with a+b=m . -
Finite Geometries
Finite Geometries Gy¨orgyKiss June 26th, 2012, Rogla gyk Finite Geometries Chain of length h : x0 I x1 I ::: I xh where xi 2 P [ L: The distance of two elements d(x; y) : length of the shortest chain joining them. Generalized polygons Let S = (P; L; I) be a connected, finite point-line incidence geometry. P and L are two distinct sets, the elements of P are called points, the elements of L are called lines. I ⊂ (P × L) [ (L × P) is a symmetric relation, called incidence. gyk Finite Geometries The distance of two elements d(x; y) : length of the shortest chain joining them. Generalized polygons Let S = (P; L; I) be a connected, finite point-line incidence geometry. P and L are two distinct sets, the elements of P are called points, the elements of L are called lines. I ⊂ (P × L) [ (L × P) is a symmetric relation, called incidence. Chain of length h : x0 I x1 I ::: I xh where xi 2 P [ L: gyk Finite Geometries Generalized polygons Let S = (P; L; I) be a connected, finite point-line incidence geometry. P and L are two distinct sets, the elements of P are called points, the elements of L are called lines. I ⊂ (P × L) [ (L × P) is a symmetric relation, called incidence. Chain of length h : x0 I x1 I ::: I xh where xi 2 P [ L: The distance of two elements d(x; y) : length of the shortest chain joining them. gyk Finite Geometries Generalized polygons Definition Let n > 1 be a positive integer. -
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. -
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 `. -
Self-Dual Configurations and Regular Graphs
SELF-DUAL CONFIGURATIONS AND REGULAR GRAPHS H. S. M. COXETER 1. Introduction. A configuration (mci ni) is a set of m points and n lines in a plane, with d of the points on each line and c of the lines through each point; thus cm = dn. Those permutations which pre serve incidences form a group, "the group of the configuration." If m — n, and consequently c = d, the group may include not only sym metries which permute the points among themselves but also reci procities which interchange points and lines in accordance with the principle of duality. The configuration is then "self-dual," and its symbol («<*, n<j) is conveniently abbreviated to na. We shall use the same symbol for the analogous concept of a configuration in three dimensions, consisting of n points lying by d's in n planes, d through each point. With any configuration we can associate a diagram called the Menger graph [13, p. 28],x in which the points are represented by dots or "nodes," two of which are joined by an arc or "branch" when ever the corresponding two points are on a line of the configuration. Unfortunately, however, it often happens that two different con figurations have the same Menger graph. The present address is concerned with another kind of diagram, which represents the con figuration uniquely. In this Levi graph [32, p. 5], we represent the points and lines (or planes) of the configuration by dots of two colors, say "red nodes" and "blue nodes," with the rule that two nodes differently colored are joined whenever the corresponding elements of the configuration are incident. -
Algorithms and Computation in Mathematics • Volume 15
Algorithms and Computation in Mathematics • Volume 15 Editors Arjeh M. Cohen Henri Cohen David Eisenbud Bernd Sturmfels Petteri Kaski Patric R.J. Östergård Classification Algorithms for Codes and Designs With 61 Figures and 30 Tables ABC Authors Petteri Kaski Department of Computer Science and Engineering Helsinki University of Technology P. O. Box 5400 2015 HUT, Helsinki Finland e-mail: petteri.kaski@hut.fi Patric R.J. Östergård Department of Electrical and Communications, Engineering Helsinki University of Technology P.O. Box 3000 2015 HUT, Helsinki Finland e-mail: patric.ostergard@hut.fi Library of Congress Control Number: 2005935445 Mathematics Subject Classification (2000): 05-02, 05Bxx, 05Cxx, 05E20, 51Exx, 68-02, 68Rxx, 94-02, 94Bxx ISSN 1431-1550 ISBN-10 3-540-28990-9 Springer Berlin Heidelberg New York ISBN-13 978-3-540-28990-6 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. -
1 Turán's Theorem
Yuval Wigderson Turandotdotdot January 27, 2020 Ma il mio mistero `echiuso in me Nessun dorma, from Puccini's Turandot Libretto: G. Adami and R. Simoni 1 Tur´an'stheorem How many edges can we place among n vertices in such a way that we make no triangle? This is perhaps the first question asked in the field of extremal graph theory, which is the topic of this talk. Upon some experimentation, one can make the conjecture that the best thing to do is to split the vertices into two classes, of sizes x and n − x, and then connect any two vertices in different classes. This will not contain a triangle, since by the pigeonhole principle, among any three vertices there will be two in the same class, which will not be adjacent. This construction will have x(n − x) edges, and by the AM-GM inequality, we see that this is maximized when x and n − x are as close as possible; namely, our classes should n n n n n2 1 n have sizes b 2 c and d 2 e. Thus, this graph will have b 2 cd 2 e ≈ 4 ≈ 2 2 edges. Indeed, this is the unique optimal construction, as was proved by Mantel in 1907. In 1941, Tur´anconsidered a natural extension of this problem, where we forbid not a triangle, but instead a larger complete graph Kr+1. As before, a natural construction is to split the vertices into r almost-equal classes and connect all pairs of vertices in different classes. Definition 1.