Handbook of the Tutte Polynomial and Related Topics

805 Pages·2022·36.564 MB·English

by  Joanna A. Ellis-Monaghan, Iain Moffatt

Handbook of the
Tutte Polynomial and
Related Topics

Handbook of the
Tutte Polynomial and
Related Topics
Edited by
Joanna A. Ellis-Monaghan
Korteweg - de Vries Instituut voor Wiskunde, Universiteit van
Amsterdam, Netherlands
Iain Moffatt
Royal Holloway, University of London, United Kingdom
First edition published 2022
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
© 2022 Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, LLC
Reasonable efforts have been made to publish reliable data and information, but the author and
publisher cannot assume responsibility for the validity of all materials or the consequences of their use.
The authors and publishers have attempted to trace the copyright holders of all material reproduced in
this publication and apologize to copyright holders if permission to publish in this form has not been
obtained. If any copyright material has not been acknowledged please write and let us know so we may
rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information storage
or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, access www.copyright.com
or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-
750-8400. For works that are not available on CCC please contact [email protected]
Trademark notice: Product or corporate names may be trademarks or registered trademarks and are
used only for identification and explanation without intent to infringe.
Library of Congress Cataloging‑in‑Publication Data
Names: Ellis-Monaghan, Joanna Anthony, editor. | Moffatt, Iain, editor.
Title: Handbook of the Tutte polynomial and related topics / edited by
Joanna A. Ellis-Monaghan, Iain Moffatt.
Description: First edition. | Boca Raton : C&H/CRC Press, 2022. | Includes
bibliographical references and index.
Identifiers: LCCN 2021048725 (print) | LCCN 2021048726 (ebook) | ISBN
9781482240627 (hardback) | ISBN 9781032231938 (paperback) | ISBN
9780429161612 (ebook)
Subjects: LCSH: Tutte polynomial. | Graph theory. | Polynomials. |
Invariants.
Classification: LCC QA166.249 .H36 2022 (print) | LCC QA166.249 (ebook) |
DDC 511/.5--dc23/eng/20211223
LC record available at https://lccn.loc.gov/2021048725
LC ebook record available at https://lccn.loc.gov/2021048726
ISBN: 9781482240627 (hbk)
ISBN: 9781032231938 (pbk)
ISBN: 9780429161612 (ebk)
DOI: 10.1201/9780429161612
Typeset in CMR10
by KnowledgeWorks Global Ltd.
Publisher's note: This book has been prepared from camera-ready copy provided by the authors.
Contents
Preface xv
Contributors xvii
I Fundamentals 1
1 Graph theory 3
Joanna A. Ellis-Monaghan • Iain Moffatt
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Graph theory conventions . . . . . . . . . . . . . . . . . . . 3
2 The Tutte polynomial for graphs 14
Joanna A. Ellis-Monaghan • Iain Moffatt
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 The standard definitions of the Tutte polynomial . . . . . . 15
2.3 Multiplicativity . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Universality of the Tutte polynomial . . . . . . . . . . . . . 24
2.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Essential properties of the Tutte polynomial 27
B´ela Bollob´as • Oliver Riordan
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Evaluations at special points . . . . . . . . . . . . . . . . . 28
3.3 Coefficient properties and irreducibility . . . . . . . . . . . . 32
3.4 Evaluations along curves . . . . . . . . . . . . . . . . . . . . 34
3.5 Dual graphs and medial graphs . . . . . . . . . . . . . . . . 36
3.6 Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.7 Signed, colored and topological Tutte polynomials . . . . . 42
3.8 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Matroid theory 44
James Oxley
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Fundamental examples and definitions . . . . . . . . . . . . 45
4.3 The many faces of a matroid . . . . . . . . . . . . . . . . . 51
4.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
v
vi Contents
4.5 Basic constructions . . . . . . . . . . . . . . . . . . . . . . . 60
4.6 More examples . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7 The Tutte polynomial of a matroid . . . . . . . . . . . . . . 73
4.8 Some particular evaluations . . . . . . . . . . . . . . . . . . 81
4.9 Some basic identities . . . . . . . . . . . . . . . . . . . . . . 85
5 Tutte polynomial activities 86
Spencer Backman
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Activities for maximal spanning forests . . . . . . . . . . . . 87
5.3 Activity bipartition . . . . . . . . . . . . . . . . . . . . . . . 88
5.4 Activities for subgraphs . . . . . . . . . . . . . . . . . . . . 89
5.5 Depth-first search external activity . . . . . . . . . . . . . . 90
5.6 Activities via combinatorial maps . . . . . . . . . . . . . . . 91
5.7 Unified activities for subgraphs via decision trees . . . . . . 93
5.8 Orientation activities . . . . . . . . . . . . . . . . . . . . . . 94
5.9 Active orders . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.10 Shellability and activity . . . . . . . . . . . . . . . . . . . . 97
5.11 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Tutte uniqueness and Tutte equivalence 100
Joseph E. Bonin • Anna de Mier
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2 Basic notions and results, and initial examples . . . . . . . 101
6.3 Tutte uniqueness and equivalence for graphs . . . . . . . . . 108
6.4 Tutte uniqueness and equivalence for matroids . . . . . . . 116
6.5 Related results . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.6 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 137
II Computation 139
7 Computational techniques 141
Criel Merino
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.2 Direct computation . . . . . . . . . . . . . . . . . . . . . . . 142
7.3 Duality and matroid operations . . . . . . . . . . . . . . . . 145
7.4 Using equivalent polynomials . . . . . . . . . . . . . . . . . 147
7.5 Transfer matrix method . . . . . . . . . . . . . . . . . . . . 151
7.6 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.7 Counting arguments . . . . . . . . . . . . . . . . . . . . . . 156
7.8 Other strategies . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.9 Computation for common graphs and matroids . . . . . . . 159
7.10 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 159
Contents vii
8 Computational resources 161
David Pearce • Gordon F. Royle
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.2 Implementations . . . . . . . . . . . . . . . . . . . . . . . . 162
8.3 Comparative performance . . . . . . . . . . . . . . . . . . . 170
8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
8.5 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 173
9 The exact complexity of the Tutte polynomial 175
Tomer Kotek • Johann A. Makowsky
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9.2 Complexity classes and graph width . . . . . . . . . . . . . 176
9.3 Exact evaluations on graphs . . . . . . . . . . . . . . . . . . 179
9.4 Exact evaluation on matroids . . . . . . . . . . . . . . . . . 184
9.5 Algebraic models of computation . . . . . . . . . . . . . . . 190
9.6 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 192
10 Approximating the Tutte polynomial 194
Magnus Bordewich
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
10.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
10.3 Randomized approximation schemes . . . . . . . . . . . . . 196
10.4 Positive approximation results . . . . . . . . . . . . . . . . . 199
10.5 Negative results: inapproximability . . . . . . . . . . . . . . 203
10.6 The quantum connection . . . . . . . . . . . . . . . . . . . . 208
10.7 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 209
III Specializations 211
11 Foundations of the chromatic polynomial 213
Fengming Dong • Khee Meng Koh
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
11.2 Computing chromatic polynomials . . . . . . . . . . . . . . 214
11.3 Properties of chromatic polynomials . . . . . . . . . . . . . 220
11.4 Chromatically equivalent graphs . . . . . . . . . . . . . . . 232
11.5 Roots of chromatic polynomials . . . . . . . . . . . . . . . . 241
11.6 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 247
12 Flows and colorings 252
Delia Garijo • Andrew Goodall • Jaroslav Neˇsetˇril
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
12.2 Flows and the flow polynomial . . . . . . . . . . . . . . . . 253
12.3 Coloring-flow convolution formulas . . . . . . . . . . . . . . 257
12.4 A-bicycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
12.5 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 263
viii Contents
13 Skein polynomials and the Tutte polynomial when x=y 266
Joanna A. Ellis-Monaghan • Iain Moffatt
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
13.2 Vertex states, graph states, and skein relations . . . . . . . 267
13.3 Skein polynomials . . . . . . . . . . . . . . . . . . . . . . . . 269
13.4 Evaluations of the Tutte polynomial along x=y . . . . . . 280
13.5 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 283
14 The interlace polynomial and the Tutte–Martin polynomial 284
Robert Brijder • Hendrik Jan Hoogeboom
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
14.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
14.3 Interlace polynomial . . . . . . . . . . . . . . . . . . . . . . 286
14.4 Martin polynomial . . . . . . . . . . . . . . . . . . . . . . . 291
14.5 Global interlace polynomial . . . . . . . . . . . . . . . . . . 292
14.6 Two-variable interlace polynomial . . . . . . . . . . . . . . . 294
14.7 Weighted interlace polynomial . . . . . . . . . . . . . . . . . 295
14.8 Connection with the Tutte polynomial . . . . . . . . . . . . 296
14.9 Isotropic systems and the Tutte–Martin polynomial . . . . . 297
14.10 Interlace polynomials for delta-matroids . . . . . . . . . . . 298
14.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
14.12 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 303
IV Applications 305
15 Network reliability 307
Jason I. Brown • Charles J. Colbourn
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
15.2 Forms of the reliability polynomial . . . . . . . . . . . . . . 310
15.3 Calculating coefficients . . . . . . . . . . . . . . . . . . . . . 314
15.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 315
15.5 Coefficient inequalities . . . . . . . . . . . . . . . . . . . . . 316
15.6 Analytic properties of reliability . . . . . . . . . . . . . . . . 321
15.7 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 324
16 Codes 328
Thomas Britz • Peter J. Cameron
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
16.2 Linear codes and the Tutte polynomial . . . . . . . . . . . . 329
16.3 Ordered tuples and subcodes . . . . . . . . . . . . . . . . . 334
16.4 Equivalence of the Tutte polynomial and weights . . . . . . 337
16.5 Orbital polynomials . . . . . . . . . . . . . . . . . . . . . . 337
16.6 Other generalizations and applications . . . . . . . . . . . . 339
16.7 Wei’s duality theorem . . . . . . . . . . . . . . . . . . . . . 340
16.8 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 343
Contents ix
17 The chip-firing game and the sandpile model 345
Criel Merino
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
17.2 The Tutte polynomial and the chip-firing game . . . . . . . 346
17.3 Sandpile models . . . . . . . . . . . . . . . . . . . . . . . . . 348
17.4 The critical group and parking functions . . . . . . . . . . . 349
17.5 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 351
18 The Tutte polynomial and knot theory 352
Stephen Huggett
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
18.2 Graphs and links . . . . . . . . . . . . . . . . . . . . . . . . 353
18.3 Polynomial invariants of links . . . . . . . . . . . . . . . . . 355
18.4 The Tutte and Jones polynomials . . . . . . . . . . . . . . . 357
18.5 The Tutte and HOMFLYPT polynomials . . . . . . . . . . 359
18.6 Applications in knot theory . . . . . . . . . . . . . . . . . . 360
18.7 The Bollob´as–Riordan polynomial . . . . . . . . . . . . . . 362
18.8 Categorification . . . . . . . . . . . . . . . . . . . . . . . . . 364
18.9 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 366
19 Quantum field theory connections 368
Adrian Tanasa
19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
19.2 The Symanzik polynomials as evaluations of the multivariate
Tutte polynomial . . . . . . . . . . . . . . . . . . . . . . . . 369
19.3 The Symanzik polynomials in quantum field theory . . . . . 370
19.4 Hopf algebras and the Tutte polynomial . . . . . . . . . . . 375
19.5 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 377
20 The Potts and random-cluster models 378
Geoffrey Grimmett
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
20.2 Probabilistic models from physics . . . . . . . . . . . . . . . 379
20.3 Phase transition . . . . . . . . . . . . . . . . . . . . . . . . 385
20.4 Basic properties of random-cluster measures . . . . . . . . . 386
20.5 The Limit as q ↓0 . . . . . . . . . . . . . . . . . . . . . . . 387
20.6 Flow polynomial . . . . . . . . . . . . . . . . . . . . . . . . 390
20.7 The limit of zero temperature . . . . . . . . . . . . . . . . . 391
20.8 The random-cluster model on the complete graph . . . . . . 392
20.9 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 393
21 Where Tutte and Holant meet: a view from counting com-
plexity 395
Jin-Yi Cai • Tyson Williams
21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 395