Graphs and Cubes PDF
Preview Graphs and Cubes
Universitext Universitext Series Editors: Sheldon Axler San Francisco State University Vincenzo Capasso Università degli Studi di Milano Carles Casacuberta Universitat de Barcelona Angus J. MacIntyre Queen Mary, University of London Kenneth Ribet University of California, Berkeley Claude Sabbah CNRS, École Polytechnique Endre Süli University of Oxford Wojbor A. Woyczynski Case Western Reserve University Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, to very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext. For further volumes: http://www.springer.com/series/223 Sergei Ovchinnikov Graphs and Cubes Sergei Ovchinnikov Department of Mathematics San Francisco State University San Francisco, CA 94132 USA [email protected] ISSN 0172-5939 e-ISSN 2191-6675 ISBN 978-1-4614-0796-6 e-ISBN 978-1-4614-0797-3 DOI 10 .1007/978-1-4614-0797-3 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011935732 Mathematics Subject Classification (2010): 05CXX, 68R10 © Springer Science(cid:0) +Business Media, LLC 2011 All rights reserved. Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper Springer is part of Springer Science+Business Media (www.springer.com) To my dear wife, Galya Preface This book is an introductory text in graph theory, focusing on partial cubes, that is, graphs that are isometricallyembeddable intohypercubes of an arbi- trarydimension.Thisbranchofgraphtheoryhasdevelopedrapidlyduringthe past three decades, producing exciting results and establishing links to other branches of mathematics. Because of their rich structural properties, partial cubes have foundapplicationsin theoretical computer science, coding theory, data transmission, genetics, and even the political and social sciences. How- ever,thisresearchareahaspreviouslyfailedtotrickledownintograduate-level teaching of graph theory. In fact, even the term “partial cube”can’t be found instandardtextbooks ongraphtheory.Inthisbook,Iattempttoremedythis situation. Exercisingaconcrete approachtographtheory,thisbookfocuses onthree classes of graphs: bipartite graphs, cubical graphs, and partial cubes (intro- duced in Chapters 2, 4, and 5, respectively). Cubical graphs are graphs that are embeddable into hypercubes; if they are isometrically embeddable into hypercubes, then they are called partial cubes. Cubicalgraphtheory isabranch ofgraphtheory thatisreasonablysmall, yet deep enough to demonstrate the power and tools of the general theory. It can serve as a launching pad for studies of other topics in graph theory and their applications. The book is organized into eight chapters. The first four provide the con- cepts and tools needed to understand Chapter 5 (Partial Cubes), and the remainingthree chapters buildonthisunderstanding withexamples,applica- tions, and a foundationfor further exploration. The specific topics addressed in each chapter are as follows. Chapter 1 introduces several basic concepts of graph theory that are used throughout the book. I adopt the terminology and notations used by J. A. Bondy and U. S. R. Murty in their influential text Graph Theory with Applications (Bondy and Murty, 1976) and in their recent book Graph The- ory (Bondy and Murty, 2008). Many other fundamental concepts of graph theory are introduced graduallyin the rest of the book, as needed. VII VIII Preface Because allcubicalgraphsarebipartite,Chapter2presents anelementary theory of bipartite graphs. This chapter establishes various characterizations ofbipartitegraphsanddiscusses theirstructural properties.Inthediscussion, I emphasize the importance of geometric structures of betweenness and con- vexity,concepts thatrecur frequently inthe treatment ofcubicalgraphs later in the book. Chapter 3 focuses on the beautiful geometric and combinatorial objects knownashypercubes(orcubes,asIgenerallycallthemforthesakeofsimplic- ity). The first five sections of Chapter 3 consider various instances of cubes in geometry, algebra, and graph theory. Section 3.6 explores the subtleties of how a cube’s dimensionality affects its classification as a Cartesian prod- uct(finite-dimensionalcubes are Cartesianproducts, butinfinite-dimensional cubes are not; they are weak Cartesian products), whereas later sections ad- dress the nature ofcubes as highlysymmetricalobjects. Section 3.8describes symmetry groups, and Section 3.10 characterizes finite cubes. Chapter 4 presents practically everything that is known about cubical graphs(andwe donotknowmuch).Unlikewithbipartitegraphs(Chapter 2) andpartialcubes (Chapter 5),there isnoeffective characterization ofcubical graphs in general. However, a criterion based on c-valuations (Section 4.2) is a useful tool for establishing properties of some special classes of cubical graphs, such as dichotomic trees (Section 4.3). Chapter 5, the central chapter of the book, presents the concept of the partial cube as a cubical graph that admits an isometric embedding into a cube. This chapter deals mainly with structural properties of partial cubes, usingtechniques introducedinChapters 2through4.Onegoalofthischapter is to establish several characterizations of partial cubes. Another goal is to demonstratehowgeneralmathematicaltechniquesofconstructingnewobjects from old ones work for partial cubes. Chapter6expands theunderstanding ofpartialcubes bydefiningthemas graphsisometricallyembeddableintointegerlattices (grids),whicharethem- selves definable as partial cubes. This chapter is devoted entirely to lattices and their isometric subgraphs. Chapter7movesontoaparticularlybeautifulgeometricexampleofpartial cubes: region graphs of hyperplane arrangements. In this chapter, I prove that such graphs are indeed partial cubes, and I present their algebraic and geometric applications. Finally,Chapter 8laysdownamathematicalfoundationforcubicalgraph applications. Two kinds of token systems—cubical systems and media—are presentedanddefinedaxiomaticallybyimposingcompellingindependentcon- ditionsonthesystems.Thelastsectionprovidesastochasticmodelforsystem evolution. Therearefewprerequisites forthemaintext.Itisassumedthatthereader is familiar with basic mathematical concepts and methods on the level of undergraduate courses in discrete mathematics, linear algebra, group theory, and topology of Euclidean spaces. Although the book is intended for lower- Preface IX division graduate students, I believe that it will find readership in a much wider audience. I have chosen a very geometric mode of presentation for this book, in accordance with its topic. The reader will find many drawings illustrating concepts, proofs, and the exercises included (along with historical notes) at the end of every chapter. I encourage readers to explore the exercises fully, and even to use them as the basis for research projects. IwanttothankmySpringereditorKaitlinLeachforhersupportthrough- out the preparation of this book. Berkeley, California Sergei Ovchinnikov June 2011
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