topics in topological graph theory

Topics in Topological Graph Theory

The use of topological ideas to explore various aspects of graph theory, and vice versa, is a fruitfularea of research. There are links with other areas of mathematics, such as design theory and geometry,and increasingly with such areas as computer networks where symmetry is an important feature.Other books cover portions of the material here, but there are no other books with such a wide scope.

This book contains fifteen expository chapters written by acknowledged international experts inthe field. Their well-written contributions have been carefully edited to enhance readability and tostandardize the chapter structure, terminology and notation throughout the book. To help the reader,there is an extensive introductory chapter that covers the basic background material in graph theoryand the topology of surfaces. Each chapter concludes with an extensive list of references.

lowell w. beineke is Schrey Professor of Mathematics at Indiana University–Purdue UniversityFort Wayne, where he has been since receiving his Ph.D. from the University of Michigan under theguidance of Frank Harary. His graph theory interests are broad, and include topological graph theory,line graphs, tournaments, decompositions and vulnerability. With Robin Wilson he edited SelectedTopics in Graph Theory (3 volumes), Applications of Graph Theory, Graph Connections and Topicsin Algebraic Graph Theory. Until recently he was editor of the College Mathematics Journal.

robin j. wilson is Professor of Pure Mathematics at The Open University, UK, and EmeritusProfessor of Geometry at Gresham College, London. After graduating from Oxford, he received hisPh.D. in number theory from the University of Pennsylvania. He has written and edited many bookson graph theory and the history of mathematics, including Introduction to Graph Theory and FourColours Suffice, and his research interests include graph colourings and the history of combinatorics.He has won a Lester Ford Award and a George Polya Award from the MAA for his expository writing.

jonathan l. gross, Professor of Computer Science at Columbia University, served as anacademic consultant for this volume. His mathematical work in topology and graph theory haveearned him an Alfred P. Sloan Fellowship, an IBM Postdoctoral Fellowship, and numerous researchgrants. With Thomas Tucker, he wrote Topological Graph Theory and several fundamentalpioneering papers on voltage graphs and on enumerative methods. He has written and edited eightbooks on graph theory and combinatorics, seven books on computer programming topics, and onebook on cultural sociometry.

thomas w. tucker, Charles Hetherington Professor of Mathematics at Colgate University, alsoserved as an academic consultant for this volume. He has been at Colgate University since 1973, aftera Ph.D. in 3-manifolds from Dartmouth in 1971 and a post-doctoral position at Princeton. He isco-author (with Jonathan Gross) of Topological Graph Theory. His early publications were onnon-compact 3-manifolds, then topological graph theory, but his recent work is mostly algebraic,especially distinguishability and the group-theoretic structure of symmetric maps.

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68 R. Goodman and N. R. Wallach Representations and Invariants of the Classical Groups69 T. Beth, D. Jungnickel, and H. Lenz Design Theory I, 2nd edn70 A. Pietsch and J. Wenzel Orthonormal Systems for Banach Space Geometry71 G. E. Andrews, R. Askey and R. Roy Special Functions72 R. Ticciati Quantum Field Theory for Mathematicians73 M. Stern Semimodular Lattices74 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations I75 I. Lasiecka and R. Triggiani Control Theory for Partial Differential Equations II76 A. A. Ivanov Geometry of Sporadic Groups I77 A. Schinzel Polynomials with Special Regard to Reducibility78 T. Beth, D. Jungnickel and H. Lenz Design Theory II, 2nd edn79 T. W. Palmer Banach Algebras and the General Theory of *-Albegras II80 O. Stormark Lie’s Structural Approach to PDE Systems81 C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables82 J. P. Mayberry The Foundations of Mathematics in the Theory of Sets83 C. Foias, O. Manley, R. Rosa and R. Temam Navier–Stokes Equations and Turbulence84 B. Polster and G. Steinke Geometries on Surfaces85 R. B. Paris and D. Kaminski Asymptotics and Mellin–Barnes Integrals86 R. McEliece The Theory of Information and Coding, 2nd edn87 B. A. Magurn An Algebraic Introduction to K-Theory88 T. Mora Solving Polynomial Equation Systems I89 K. Bichteler Stochastic Integration with Jumps90 M. Lothaire Algebraic Combinatorics on Words91 A. A. Ivanov and S. V. Shpectorov Geometry of Sporadic Groups II92 P. McMullen and E. Schulte Abstract Regular Polytopes93 G. Gierz et al. Continuous Lattices and Domains94 S. R. Finch Mathematical Constants95 Y. Jabri The Mountain Pass Theorem96 G. Gasper and M. Rahman Basic Hypergeometric Series, 2nd edn97 M. C. Pedicchio and W. Tholen (eds.) Categorical Foundations98 M. E. H. Ismail Classical and Quantum Orthogonal Polynomials in One Variable99 T. Mora Solving Polynomial Equation Systems II

100 E. Olivieri and M. Eulalia Vares Large Deviations and Metastability101 A. Kushner, V. Lychagin and V. Rubtsov Contact Geometry and Nonlinear Differential Equations102 L. W. Beineke and R. J. Wilson (eds.) with P. J. Cameron Topics in Algebraic Graph Theory103 O. Staffans Well-Posed Linear Systems104 J. M. Lewis, S. Lakshmivarahan and S. K. Dhall Dynamic Data Assimilation105 M. Lothaire Applied Combinatorics on Words106 A. Markoe Analytic Tomography107 P. A. Martin Multiple Scattering108 R. A. Brualdi Combinatorial Matrix Classes110 M.-J. Lai and L. L. Schumaker Spline Functions on Triangulations111 R. T. Curtis Symmetric Generation of Groups112 H. Salzmann, T. Grundhofer, H. Hahl and R. Lowen The Classical Fields113 S. Peszat and J. Zabczyk Stochastic Partial Differential Equations with Levy Noise114 J. Beck Combinatorial Games116 D. Z. Arov and H. Dym J-Contractive Matrix Valued Functions and Related Topics117 R. Glowinski, J.-L. Lions and J. He Exact and Approximate Controllability for Distributed Parameter

Systems118 A. A. Borovkov and K. A. Borovkov Asymptotic Analysis of Random Walks119 M. Deza and M. Dutour Sikiric Geometry of Chemical Graphs120 T. Nishiura Absolute Measurable Spaces121 M. Prest Purity, Spectra and Localisation122 S. Khrushchev Orthogonal Polynomials and Continued Fractions: From Euler’s Point of View123 H. Nagamochi and T. Ibaraki Algorithmic Aspects of Graph Connectivity124 F. W. King Hilbert Transforms I125 F. W. King Hilbert Transforms II126 O. Calin and D.-C. Chang Sub-Riemannian Geometry127 M. Grabisch, J.-L. Marichal, R. Mesiar and E. Pap Aggregation Functions






Leonhard Euler (1707–1783),

the founder of topological graph theory.






Topics in Topological Graph Theory

Edited by

LOWELL W. BEINEKEIndiana University–Purdue University

Fort Wayne

ROBIN J. WILSONThe Open University

Academic Consultants

JONATHAN L. GROSSColumbia University

THOMAS W. TUCKERColgate University






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c© Cambridge University Press 2009

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First published 2009

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This book is dedicated to the memory of Gerhard Ringel (1919–2008),one of the pioneers of modern topological graph theory.






Contents

Foreword by Jonathan L. Gross and Thomas W. Tucker page xvPreface xvii

Introduction 1LOWELL W. BEINEKE and ROBIN J. WILSON

1. Graph theory 12. Graphs in the plane 103. Surfaces 124. Graphs on surfaces 14

1 Embedding graphs on surfaces 18JONATHAN L. GROSS and THOMAS W. TUCKER

1. Introduction 182. Graphs and surfaces 193. Embeddings 204. Rotation systems 235. Covering spaces and voltage graphs 266. Enumeration 297. Algorithms 308. Graph minors 31

2 Maximum genus 34JIANER CHEN and YUANQIU HUANG

1. Introduction 342. Characterizations and complexity 363. Kuratowski-type theorems 384. Upper-embeddability 395. Lower bounds 40

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x Contents

3 Distribution of embeddings 45JONATHAN L. GROSS

1. Introduction 452. Enumerating embeddings by surface type 483. Total embedding distributions 514. Congruence classes 535. The unimodality problem 556. Average genus 567. Stratification of embeddings 59

4 Algorithms and obstructions for embeddings 62BOJAN MOHAR

1. Introduction 622. Planarity 643. Outerplanarity and face covers 664. Disc embeddings and the 2-path problem 685. Graph minors and obstructions 696. Algorithms for embeddability in general surfaces 737. Computing the genus 75

5 Graph minors: generalizing Kuratowski’s theorem 81R. BRUCE RICHTER

1. Introduction 812. Graph decompositions 843. Linked decompositions 884. Graphs with bounded tree-width 945. Finding large grids 996. Embedding large grids 107

6 Colouring graphs on surfaces 111JOAN P. HUTCHINSON

1. Introduction 1112. High-end colouring 1133. A transition from high-end to low-end colouring 1164. Colouring graphs with few colours 1195. Girth and chromatic number 1246. List-colouring graphs 1257. More colouring extensions 1278. An open problem 129






Contents xi

7 Crossing numbers 133R. BRUCE RICHTER and G. SALAZAR

1. Introduction 1332. What is the crossing number? 1353. General bounds 1374. Applications to geometry 1395. Crossing-critical graphs 1396. Other families of graphs 1437. Algorithmic questions 1448. Drawings in other surfaces 1469. Conclusion 147

8 Representing graphs and maps 151TOMAZ PISANSKI and ARJANA ZITNIK

1. Introduction 1512. Representations of graphs 1523. Energy and optimal representations 1554. Representations of maps 1635. Representations of maps in the plane 1706. Representations of incidence geometries and related topics 174

9 Enumerating coverings 181JIN HO KWAK and JAEUN LEE

1. Introduction 1812. Graph coverings 1833. Regular coverings 1854. Surface branched coverings 1905. Regular surface branched coverings 1936. Distribution of surface branched coverings 1957. Further remarks 196

10 Symmetric maps 199JOZEF SIRAN and THOMAS W. TUCKER

1. Introduction 1992. Representing maps algebraically 2003. Regular maps 2054. Cayley maps 2105. Regular Cayley maps 2126. Edge-transitive maps 2187. Maps and mathematics 221






xii Contents

11 The genus of a group 225THOMAS W. TUCKER

1. Introduction 225

2. Symmetric embeddings and groups acting on surfaces 226

3. Quotient embeddings and voltage graphs 228

4. Inequalities 232

5. Groups of low genus 235

6. Genera of families of groups 239

12 Embeddings and geometries 245ARTHUR T. WHITE

1. Introduction 245

2. Surface models 248

3. Projective geometries 250

4. Affine geometries 253

5. 3-configurations 256

6. Partial geometries 260

7. Regular embeddings for PG(2, n) 264

8. Problems 265

13 Embeddings and designs 268M. J. GRANNELL and T. S. GRIGGS

1. Introduction 268

2. Steiner triple systems and triangulations 270

3. Recursive constructions 273

4. Small systems 278

5. Cyclic embeddings 280

6. Concluding remarks 284

14 Infinite graphs and planar maps 289MARK E. WATKINS

1. Introduction 289

2. Ends 290

3. Automorphisms 293

4. Connectivities 295

5. Growth 300

6. Infinite planar graphs and maps 303






Contents xiii

15 Open problems 313DAN ARCHDEACON

1. Introduction 3132. Drawings and crossings 3143. Genus and obstructions 3174. Cycles and factors 3205. Colourings and flows 3226. Local planarity 3247. Thickness, book embeddings and covering graphs 3258. Geometrical topics 3289. Algorithms 330

10. Infinite graphs 332

Notes on contributors 337Index 341






Foreword

The origins of topological graph theory lie in the 19th century, largely with thefour colour problem and its extension to higher-order surfaces – the Heawoodmap problem. With the explosive growth of topology in the early 20th century,mathematicians like Veblen, Rado and Papakyriakopoulos provided foundationalresults for understanding surfaces combinatorially and algebraically. Kuratowski,MacLane and Whitney in the 1930s approached the four colour problem as aquestion about the structure of graphs that can be drawn without edge-crossingsin the plane. Kuratowski’s theorem characterizing planarity by two obstructionsis the most famous, and its generalization to the higher-order surfaces became aninfluential unsolved problem.

The second half of the 20th century saw the solutions of all three problems: theHeawood map problem by Ringel, Youngs et al. by 1968, the four colour problemby Appel and Haken in 1976, and finally the generalized Kuratowski problem byRobertson and Seymour in the mid-1990s. Each is a landmark of 20th-centurymathematics. The Ringel–Youngs work led to an alliance between combinatoricsand the algebraic topology of branched coverings. The Appel–Haken work was thefirst time that a mathematical theorem relied on exhaustive computer calculations.And the Robertson–Seymour work led to their solution of Wagner’s conjecture,which provides a breathtaking structure for the collection of all finite graphs, acollection that would seem to have no structure at all.

Each of these problems centres on the question of which graphs can beembedded in which surfaces, with two complementary perspectives – fixing thegraph or fixing the surface. Although the question sounds highly focused, the studyof graphs on surfaces turns out to be incredibly broad, rich in connections with otherbranches of mathematics and computer science: algorithms, computer-drawing,group theory, Riemann surfaces, enumerative combinatorics, block designs, finitegeometries, Euclidean and non-Euclidean geometry, knot theory, the absoluteGalois group, C*-algebras, and even string theory.

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xvi Foreword

This volume attempts to survey the principal results within over-arching themesfor the myriad aspects of topological graph theory. The authors of the chapters arerecognized authorities in their fields. This book is written for the non-specialist andcan be used as the basis for a graduate-level course. Nonetheless, the individualchapters cover their fields in great depth and detail, so that even specialists will findthe book valuable, both as a reference and as a source of new insights and problems.

JONATHAN L. GROSSTHOMAS W. TUCKER






Preface

The field of graph theory has undergone tremendous growth during the past century.As recently as fifty years ago, the graph theory community had few membersand most were in Europe and North America; today there are hundreds of graphtheorists and they span the globe. By the mid-1970s, the field had reached thepoint where we perceived the need for a collection of surveys of the areas ofgraph theory: the result was our three-volume series Selected Topics in GraphTheory, comprising articles written by distinguished experts in a common style.During the past quarter-century, the transformation of the subject has continued,with individual areas (such as topological graph theory) expanding to the point ofhaving important sub-branches themselves. This inspired us to conceive of a newseries of books, each a collection of articles within a particular area written byexperts within that area. The first of these books was our companion volume onalgebraic graph theory, published in 2004. This is the second of these books.

One innovative feature of these volumes is the engagement of academicconsultants (here, Jonathan Gross and Thomas Tucker) to advise us on topicsto be included and authors to be invited. We believe that this has been successful,the result being chapters covering the full range of areas within topological graphtheory written by authors from around the world. Another important feature is thatwe have imposed uniform terminology and notation throughout, as far as possible,in the belief that this will aid readers in going from one chapter to another. For asimilar reason we have not attempted to remove a small amount of overlap betweenthe various chapters.

We hope that these features will make the book easier to use in an advancedcourse or seminar. We heartily thank the authors for cooperating on this,even though it sometimes required their abandoning some of their favouriteconventions – for example, many mathematicians use χ to denote the Eulercharacteristic, whereas for graph theorists χ usually denotes the chromatic number:the graph theorists won on this one. We also asked our contributors to undergo

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xviii Preface

the ordeal of having their early versions subjected to detailed critical reading. Webelieve that the final product is thereby significantly better than it might otherwisehave been, simply a collection of individually authored chapters. We want toexpress our sincere appreciation to all of our contributors for their cooperation.

We extend special thanks to Jonathan Gross and Thomas Tucker for theirwillingness to share their expertise as academic consultants – their advice has beeninvaluable. We are also grateful to Cambridge University Press for publishing thiswork; in particular, we thank Roger Astley and Clare Dennison for their advice,support and cooperation. Finally, we extend our appreciation to several universitiesfor the different ways in which they have assisted with this endeavour: the firsteditor is grateful to Indiana University–Purdue University in Fort Wayne, while thesecond editor has had the cooperation of the Open University and Keble College,Oxford.

LOWELL W. BEINEKEROBIN J. WILSON






topics in topological graph theory

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