Splittable and Unsplittable Graphs and Configurations
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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. -
Drawing Graphs and Maps with Curves
Report from Dagstuhl Seminar 13151 Drawing Graphs and Maps with Curves Edited by Stephen Kobourov1, Martin Nöllenburg2, and Monique Teillaud3 1 University of Arizona – Tucson, US, [email protected] 2 KIT – Karlsruhe Institute of Technology, DE, [email protected] 3 INRIA Sophia Antipolis – Méditerranée, FR, [email protected] Abstract This report documents the program and the outcomes of Dagstuhl Seminar 13151 “Drawing Graphs and Maps with Curves”. The seminar brought together 34 researchers from different areas such as graph drawing, information visualization, computational geometry, and cartography. During the seminar we started with seven overview talks on the use of curves in the different communities represented in the seminar. Abstracts of these talks are collected in this report. Six working groups formed around open research problems related to the seminar topic and we report about their findings. Finally, the seminar was accompanied by the art exhibition Bending Reality: Where Arc and Science Meet with 40 exhibits contributed by the seminar participants. Seminar 07.–12. April, 2013 – www.dagstuhl.de/13151 1998 ACM Subject Classification I.3.5 Computational Geometry and Object Modeling, G.2.2 Graph Theory, F.2.2 Nonnumerical Algorithms and Problems Keywords and phrases graph drawing, information visualization, computational cartography, computational geometry Digital Object Identifier 10.4230/DagRep.3.4.34 Edited in cooperation with Benjamin Niedermann 1 Executive Summary Stephen Kobourov Martin Nöllenburg Monique Teillaud License Creative Commons BY 3.0 Unported license © Stephen Kobourov, Martin Nöllenburg, and Monique Teillaud Graphs and networks, maps and schematic map representations are frequently used in many fields of science, humanities and the arts. -
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. -
Vertex-Transitive Graphs That Have No Hamilton Decomposition
Vertex-transitive graphs that have no Hamilton decomposition Darryn Bryant ∗ Matthew Dean † Abstract It is shown that there are infinitely many connected vertex-transitive graphs that have no Hamilton decomposition, including infinitely many Cayley graphs of valency 6, and including Cayley graphs of arbitrarily large valency. 1 Introduction A famous question of Lov´asz concerns the existence of Hamilton paths in vertex-transitive graphs [28], and no example of a connected vertex-transitive graph with no Hamilton path is known. The related question concerning the existence of Hamilton cycles in vertex-transitive graphs is another interesting and well-studied problem in graph theory, see the survey [23]. A Hamiltonian graph is a graph containing a Hamilton cycle. Thomassen (see [10, 23]) has conjectured that there are only finitely many non-Hamiltonian connected vertex-transitive graphs. On the other hand, Babai [8, 9] has conjectured that there are infinitely many such graphs. To date only five are known. These arXiv:1408.5211v3 [math.CO] 12 Nov 2014 are the complete graph of order 2, the Petersen graph, the Coxeter graph, and the two graphs obtained from the Petersen and Coxeter graphs by replacing each vertex with a triangle. For a regular graph of valency at least 4, a stronger property than the existence of a Hamilton cycle is the existence of a Hamilton decomposition. If X is a k-valent graph, then a Hamilton k decomposition of X is a set of ⌊ 2 ⌋ pairwise edge-disjoint Hamilton cycles in X. Given the small number of non-Hamiltonian connected vertex-transitive graphs, and the uncertainty concerning the existence of others, it is natural to ask how many connected vertex-transitive graphs have no Hamilton decomposition. -
Semisymmetric Graphs from Polytopes
Semisymmetric Graphs from Polytopes Barry Monson¤ University of New Brunswick Fredericton, New Brunswick, Canada E3B 5A3 ([email protected]) Toma·z Pisanskiy University of Ljubljana, FMF Jadranska 19, Ljubljana 1111, Slovenia; and Faculty of Education, University of Primorska, Cankarjeva 5, Koper 6000, Slovenia ([email protected]) Egon Schultez Northeastern University Boston MA 02115, USA ([email protected]) Asia Ivi¶c Weissx York University Toronto, Ontario, Canada M3J 1P3 ([email protected]) Abstract Every ¯nite, self-dual, regular (or chiral) 4-polytope of type f3;q;3g has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by drop- ping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope. Key Words: semisymmetric graphs, abstract regular and chiral polytopes. AMS Subject Classi¯cation (1991): Primary: 05C25. Secondary: 51M20. ¤Supported by the NSERC of Canada, grant 4818 ySupported by Ministry of Higher Education, Science and Technolgy of Slovenia grants P1-0294,J1-6062,L1-7230. zSupported by NSA-grant H98230-05-1-0027 xSupported by the NSERC of Canada, grant 8857 1 1 Introduction The theory of symmetric trivalent graphs and the theory of regular polytopes are each abundant sources of beautiful mathematical ideas. In [22], two of the authors established some general and unexpected connections between the two subjects, building upon a rich variety of examples appearing in the literature (see [4], [7], [10], [11], [28] and [29]). Here we develop these connections a little further, with speci¯c focus on semisymmetric graphs. -
Lombardi Drawings of Graphs 1 Introduction
Lombardi Drawings of Graphs Christian A. Duncan1, David Eppstein2, Michael T. Goodrich2, Stephen G. Kobourov3, and Martin Nollenburg¨ 2 1Department of Computer Science, Louisiana Tech. Univ., Ruston, Louisiana, USA 2Department of Computer Science, University of California, Irvine, California, USA 3Department of Computer Science, University of Arizona, Tucson, Arizona, USA Abstract. We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs. 1 Introduction The American artist Mark Lombardi [24] was famous for his drawings of social net- works representing conspiracy theories. Lombardi used curved arcs to represent edges, leading to a strong aesthetic quality and high readability. Inspired by this work, we intro- duce the notion of a Lombardi drawing of a graph, in which edges are drawn as circular arcs with perfect angular resolution: consecutive edges are evenly spaced around each vertex. While not all vertices have perfect angular resolution in Lombardi’s work, the even spacing of edges around vertices is clearly one of his aesthetic criteria; see Fig. 1. Traditional graph drawing methods rarely guarantee perfect angular resolution, but poor edge distribution can nevertheless lead to unreadable drawings. Additionally, while some tools provide options to draw edges as curves, most rely on straight-line edges, and it is known that maintaining good angular resolution can result in exponential draw- ing area for straight-line drawings of planar graphs [17,25]. -
The Wiener Dimension of a Graph
The Wiener dimension of a graph Yaser Alizadeh Department of Mathematics Tarbiat Modares University, Tehran, Iran [email protected] Sandi Klavˇzar∗ Faculty of Mathematics and Physics University of Ljubljana, Slovenia and Faculty of Natural Sciences and Mathematics University of Maribor, Slovenia [email protected] March 13, 2012 Abstract The Wiener dimension of a connected graph is introduced as the number of different distances of its vertices. For any integer d and any integer k, a graph of diameter d and of Wiener dimension k is constructed. An infinite family of non-vertex-transitive graphs with Wiener dimension 1 is also presented and it is proved that a graph of dimension 1 is 2-connected. Keywords: Wiener index; Cartesian product graphs; vertex-transitive graphs AMS subject classification (2010): 05C12, 05C25, 92E10 1 Introduction The distance considered in this paper is the usual shortest path distance. We assume throughout the paper that all graphs are connected unless stated otherwise. For terms not defined here we refer to [13]. Let G be a graph and u ∈ V (G). Then the distance of u is dG(u)= X dG(u, v) . v∈V (G) ∗ This work has been financed by ARRS Slovenia under the grant P1-0297 and within the EU- ROCORES Programme EUROGIGA/GReGAS of the European Science Foundation. The author is also with the Institute of Mathematics, Physics and Mechanics, Ljubljana. 1 In location theory, sets of vertices with the minimum (or maximum) distance in a graph play s special role because they form target sets for locations of facilities. -
Use the K-Neighborhood Subgraphs to Compute Canonical Labelings of Graphs
mathematics Article Use the K-Neighborhood Subgraphs to Compute Canonical Labelings of Graphs Jianqiang Hao 1,*, Yunzhan Gong 2, Jianzhi Sun 1 and Li Tan 1 1 Beijing Key Laboratory of Big Data Technology for Food Safety, Beijing Technology and Business University, No. 11, Fu Cheng Road, Beijing 100048, China 2 State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, No 10, Xitucheng Road, Haidian District, Beijing 100876, China * Correspondence: [email protected]; Tel.: +86-10-6898-5704 Received: 5 July 2019; Accepted: 27 July 2019; Published: 31 July 2019 Abstract: This paper puts forward an innovative theory and method to calculate the canonical labelings of graphs that are distinct to Nauty’s. It shows the correlation between the canonical labeling of a graph and the canonical labeling of its complement graph. It regularly examines the link between computing the canonical labeling of a graph and the canonical labeling of its open k-neighborhood subgraph . It defines dif fusion degree sequences and entire dif fusion degree sequence . For each node of a graph G, it designs a characteristic m_NearestNode to improve the precision for calculating canonical labeling. Two theorems established here display how to compute the first nodes of MaxQ(G). Another theorem presents how to determine the second nodes of MaxQ(G). When computing Cmax(G), if MaxQ(G) already holds the first i nodes u1, u2, ··· , ui, Diffusion and Nearest Node theorems provide skill on how to pick the succeeding node of MaxQ(G). Further, it also establishes two theorems to determine the Cmax(G) of disconnected graphs. -
Induced Saturation Number Jason James Smith Iowa State University
Iowa State University Digital Repository @ Iowa State University Graduate Theses and Dissertations Graduate College 2012 Induced Saturation Number Jason James Smith Iowa State University Follow this and additional works at: http://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Smith, Jason James, "Induced Saturation Number" (2012). Graduate Theses and Dissertations. Paper 12465. This Dissertation is brought to you for free and open access by the Graduate College at Digital Repository @ Iowa State University. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Digital Repository @ Iowa State University. For more information, please contact [email protected]. Induced saturation number by Jason James Smith A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Ryan Martin, Major Professor Leslie Hogben Maria Axenovich Ling Long Jack Lutz Iowa State University Ames, Iowa 2012 Copyright © Jason James Smith, 2012. All rights reserved. ii TABLE OF CONTENTS LIST OF FIGURES . iv ACKNOWLEDGEMENTS . vi ABSTRACT . vii CHAPTER 1. INTRODUCTION . 1 1.1 Saturation in Graphs . .1 1.1.1 Definitions and Notation . .2 1.1.2 Applications . .4 CHAPTER 2. INDUCED SATURATION NUMBER AND BOUNDS . 5 2.1 Bounds for Induced Saturation Number . .5 2.2 Induced Saturation Number for a Few Families of Graphs . 12 n+1 CHAPTER 3. INDUCED SATURATION NUMBER OF P4 IS 3 ..... 15 3.1 Definitions and Notation . 15 3.2 Upper Bound by Construction . 17 3.3 Lower Bound by Induction . 19 3.4 Technical Lemma . -
Math.RA] 25 Sep 2013 Previous Paper [3], Also Relying in Conceptually Separated Tools from Them, Such As Graphs and Digraphs
Certain particular families of graphicable algebras Juan Núñez, María Luisa Rodríguez-Arévalo and María Trinidad Villar Dpto. Geometría y Topología. Facultad de Matemáticas. Universidad de Sevilla. Apdo. 1160. 41080-Sevilla, Spain. [email protected] [email protected] [email protected] Abstract In this paper, we introduce some particular families of graphicable algebras obtained by following a relatively new line of research, ini- tiated previously by some of the authors. It consists of the use of certain objects of Discrete Mathematics, mainly graphs and digraphs, to facilitate the study of graphicable algebras, which are a subset of evolution algebras. 2010 Mathematics Subject Classification: 17D99; 05C20; 05C50. Keywords: Graphicable algebras; evolution algebras; graphs. Introduction The main goal of this paper is to advance in the research of a novel mathematical topic emerged not long ago, the evolution algebras in general, and the graphicable algebras (a subset of them) in particular, in order to obtain new results starting from those by Tian (see [4, 5]) and others already obtained by some of us in a arXiv:1309.6469v1 [math.RA] 25 Sep 2013 previous paper [3], also relying in conceptually separated tools from them, such as graphs and digraphs. Concretely, our goal is to find some particular types of graphicable algebras associated with well-known types of graphs. The motivation to deal with evolution algebras in general and graphicable al- gebras in particular is due to the fact that at present, the study of these algebras is very booming, due to the numerous connections between them and many other branches of Mathematics, such as Graph Theory, Group Theory, Markov pro- cesses, dynamic systems and the Theory of Knots, among others. -
On Cubic Graphs Admitting an Edge-Transitive Solvable Group
Journal of Algebraic Combinatorics, 20, 99–113, 2004 c 2004 Kluwer Academic Publishers. Manufactured in The Netherlands. On Cubic Graphs Admitting an Edge-Transitive Solvable Group ALEKSANDER MALNICˇ ∗ DRAGAN MARUSIˇ Cˇ ∗ PRIMOZˇ POTOCNIKˇ † IMFM, Oddelek za matematiko, Univerza v Ljubljani, Jadranska 19, 1111 Ljubljana, Slovenija Received March 14, 2002; Revised August 21, 2003; Accepted October 14, 2003 Abstract. Using covering graph techniques, a structural result about connected cubic simple graphs admitting an edge-transitive solvable group of automorphisms is proved. This implies, among other, that every such graph can be obtained from either the 3-dipole Dip3 or the complete graph K4,byasequence of elementary-abelian covers. Another consequence of the main structural result is that the action of an arc-transitive solvable group on a connected cubic simple graph is at most 3-arc-transitive. As an application, a new infinite family of semisymmetric cubic graphs, arising as regular elementary abelian covering projections of K3,3,isconstructed. Keywords: symmetric graph, edge transitive graph, cubic graph, trivalent graph, covering projection of graphs, solvable group of automorphisms 1. Introduction Throughout this paper graphs are assumed to be finite, and unless specified otherwise, simple, undirected and connected. It transpires that, when investigating edge-transitive cubic (simple) graphs the concept of graph coverings plays a central role. A correct treatment of this concept calls for a more general definition of a graph (see Section 2), with the class of simple graphs as a special case. The study of cubic arc-transitive graphs has its roots in the classical result of Tutte [20], who proved that cubic graphs are at most 5-arc-transitive. -
A Godsil E Thomason I Solomon M Abiad B Van Dam F Balbuena J
8.30–9.15 am A Godsil E Thomason I Solomon M Abiad 9.20–9.40 am 1 Zhan 10 Bukh 19 Srinivasan 24 Reichard 9.45–10.05 am 2 Ye 11 Martin 20 Sumalroj 25 Xu 10.05–10.35 am break break break break 10.35–10.55 am 3 Dalfó 12 Kamat 21 Bencs 26 Peng 11.00–11.20 am 4 McGinnis 13 Timmons 22 Guo 27 Kravitz 11.25–12.10 pm B Van Dam F Balbuena J Muzychuk N Xiang 12.10–2.00 pm lunch lunch lunch lunch 2.00–2.45 pm C Kantor G Füredi K Williford 2.50–3.10 pm 5 Gu 14 Tait 23 Ducey 3.15–3.35 pm 6 Coutinho 15 Kodess L Woldar break 3.40–4.00 pm 7 Greaves 16 Y. Wang 4.00–4.30 pm break break 4.30–4.50 pm 8 Fiol 17 Kronenthal 4:55–5.15 pm 9 W. Wang 18 Moorhouse 5.20–6.10 pm D Haemers H Lazebnik 2 8.30–9.15 am A Chris Godsil Spectral invariants from embeddings 9.20–9.40 am 1 Harmony Zhan Quantum walks and mixing 9.45–10.05 am 2 Dong Ye Median eigenvalues and graph inverse 10.35–10.55 am 3 Cristina Dalfó Characterizing identifying codes from the spectrum of a graph or digraph 11.00–11.20 am 4 Matt McGinnis The smallest eigenvalues of the Hamming graphs 11.25–12:10 pm B Edwin van Dam Partially metric association schemes with a small multiplicity 2.00–2.45 pm C Bill Kantor MUBs 2.50–3.10 pm 5 Xiaofeng Gu Toughness, connectivity and the spectrum of regular graphs 3.15–3.35 pm 6 Gabriel Coutinho Average mixing matrix 3.40–4.00 pm 7 Gary Greaves Edge-regular graphs and regular cliques 4.30–4.50 pm 8 Miguel Angel Fiol An algebraic approach to lifts of digraphs 4.55–5.15 pm 9 Wei Wang A positive proportion of multigraphs are determined by their generalized spectra 5.20–6.10