CPC Volume 19 Issue 5-6 Cover and Back Matter

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Zeta Functions of Graphs Networks, Crowds, Now in Paperback! A Stroll through the Garden and Markets Random Graph Dynamics AUDREY TERRAS Reasoning About a RICK DURRETT Highly Connected World Cambridge Studies in Cambridge Series in Statistical and Advanced Mathematics DAVID EASLEY Probabilistic Mathematics JON KLEINBERG $65.00: Hb: 978-0-521-11367-0: 288 pp. and “The book is written in a friendly, $50.00: Hb: 978-0-521-19533-1: 744 pp. chatty style, making it easy to read; Digital Nets and Sequences I very much like that. In summary, Random Graph Dynamics is a nice Discrepancy Theory and Permutation Patterns contribution to the area of random Quasi-Monte Carlo Integration Edited by STEVE LINTON, graphs and a source of valuable JOSEF DICK NIK RUŠKUC, insights.” and FRIEDRICH PILLICHSHAMMER and VINCENT VATTER –Journal of the $85.00: Hb: 978-0-521-19159-3: 630 pp. London Mathematical Society American Statistical Association Lecture Note Series $34.99: Pb: 978-0-521-15016-3: 220 pp. Combinatorics, Automata $75.00: Pb: 978-0-521-72834-8: 352 pp. and Number Theory An Outline of Ergodic Theory Edited by VALÉRIE BERTHÉ Boolean Models and STEVEN KALIKOW and MICHEL RIGO Methods in Mathematics, and RANDALL MCCUTCHEON Encyclopedia of Mathematics and Computer Science, and Cambridge Studies in its Applications Engineering Advanced Mathematics $59.00: Hb: 978-0-521-19440-2: 182 pp. $130.00: Hb: 978-0-521-51597-9: 636 pp. Edited by YVES CRAMA and PETER L. HAMMER P, NP, and NP-Completeness Encyclopedia of Mathematics and The Basics of Computational its Applications Complexity This collection of papers presents ODED GOLDREICH a series of in-depth examinations on a variety of advanced topics This undergraduate introduction related to Boolean functions to computational complexity and expressions. The chapters offers a wide perspective on are written by some of the two central issues in theoretical most prominent experts in computer science. The book starts their respective fields and cover with the relevant background in topics ranging from algebra computability, including Turing and propositional logic to machines, search and decision learning theory, cryptography, problems, algorithms, circuits, computational complexity, and complexity classes, and electrical engineering, and then focuses on the P-versus-NP reliability theory. Question and the theory of NP- completeness. $175.00: Hb: 978-0-521-84752-0: 780 pp. $99.00: Hb: 978-0-521-19248-4: 216 pp. $37.99: Pb: 978-0-521-12254-2

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EDITOR-IN-CHIEF SUBSCRIPTIONS BÉLA BOLLOBÁS: Trinity College, Cambridge CB2 1TQ and Department of Mathematical Sciences, University of Combinatorics, Probability and Computing (ISSN: 0963-5483) is published bimonthly in 2010 in January, March, May, July, Memphis, Memphis, TN 38152, USA, Email: [email protected] and Email: [email protected] September and November by Cambridge University Press, The Edinburgh Building, Shaftesbury Road, Cambridge, CB2 8RU, MANAGING EDITORS UK / Cambridge University Press, 32 Avenue of the Americas, New York, N.Y. 10013–2473. The subscription price of Volume 19, 2010 PAUL BALISTER: Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA, is £345 net (USA, Canada and Mexico US$640) post free, institutions print and electronic; institutions electronic only £310 / $576; Email: [email protected] single parts cost £65 net (USA, Canada and Mexico US$115) plus postage. Prices include delivery by air. Six parts form a volume. 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JEFF KAHN: Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA, Email: [email protected] Extra-combinatorial methods, hypergraphs instructions for authors YOSHIHARU KOHAYAKAWA: Instituto de Matemática e Estatística, University of São Paulo, Rua do Matão 1010, 05508-900 São Paulo, Brazil, Email: [email protected] Editorial policy The journal welcomes submissions in any of the areas of Combinatorics, Probability, or Computer Science, not just Random combinatorial structures, extremal problems those dealing explicitly with relations between the three. Its scope covers combinatorics in a broad sense, including classical and ALEXANDR KOSTOCHKA: Department of Mathematics, University of Illinois, 1409 W. 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Volume 19 Numbers 5 & 6 Sept–Nov 2010 Combinatorics, Probability & Computing Volume 19 Numbers 5 & 6 Sept–Nov 2010

CONTENTS Introduction 641 Bootstrap Percolation in High Dimensions 643 JÓZSEF BALOGH, BÉLA BOLLOBÁS AND ROBERT MORRIS Editor-in-Chief: Béla Bollobás Testing Expansion in Bounded-Degree Graphs 693 Managing Editors: Paul Balister, Imre Leader, ARTUR CZUMAJ AND CHRISTIAN SOHLER

OUE1 UBR etNv21 PAGES 641–926 2010 Sept–Nov NUMBERS 5 & 6 19 VOLUME Oliver Riordan Tight Bounds for Blind Search on the Integers and the Reals 711 MARTIN DIETZFELBINGER, JONATHAN E. ROWE, INGO WEGENER AND PHILIPP WOELFEL SPECIAL Diameters in Supercritical Random Graphs Via First Passage ISSUE Percolation 729 JIAN DING, JEONG HAN KIM, EYAL LUBETZKY AND YUVAL PERES Decompositions into Subgraphs of Small Diameter 753 JACOB FOX AND BENNY SUDAKOV Papers from the On Random Betweenness Constraints 775 2009 ANDREAS GOERDT Oberwolfach On the Number of Perfect Matchings in Random Lifts 791 CATHERINE GREENHILL, SVANTE JANSON AND ANDRZEJ RUCINSKI´ Meeting Vicarious Graphs 819 on TOMASZ LUCZAK⁄ AND ANDREW THOMASON One More Probabilistic Reformulation of the Four Colour Conjecture 829 YU. MATIYASEVICH Combinatorics The Diameter of Sparse Random Graphs 835 OLIVER RIORDAN AND NICHOLAS WORMALD and Probability

Guest Editors: Noga Alon Paul Balister Cambridge Journals Online For further information about this journal Béla Bollobás please go to the journal website at: journals.cambridge.org/cpc 0963-5483