Map Coloring, Polyhedra, and the Four-Color Problem

VOL 8 VOL DOLCIANI MATHEMATICAL EXPOSITIONS MATHEMATICAL DOLCIANI DAVID BARNETTE DAVID Map Coloring, and Polyhedra, the Four-Color Problem 178 page • 3/8” • 50lb stock • finish size: 5 1/2” x 8 1/2” • finish size: page • 3/8” • 50lb stock 178 / MAA AMS

8 Map Coloring, Polyhedra, and the Four-Color Problem BARNETTE AMS / MAA PRESS VOL 4-Color Cover DOLCIANI MATHEMATICAL EXPOSITIONS MATHEMATICAL DOLCIANI / MAA AMS 10.1090/dol/008

MAP COLORING, POLYHEDRA, AND THE FOUR-COLOR PROBLEM

By DAVID BARNETTE 'rHI~ DOLeIANI MA~rH.EMA'rI('AL EXPOSITIONS

Published hv THE MATHEMATICAL AssociA'rION OF AMERICA

Committee on Publications EDWIN F. BECKENBACH, Chairman

Subcommittee on Dolciani Mathematical Expositions ROSS HONSBERGER, Chairman D. J. ALBERS H.L.ALDER G.L.ALEXANDERSON R. P. BOAS H. EVES J. MALKEVITCH K. R. REBMAN The Dolciani Mathematical Expositions

NUMBER EIGHT

MAP COLORING, POLYHEDRA, AND THE FOUR-COLOR PROBLEM

By DAVID BARNETTE University of California, Davis

Complete Set ISBN 0-88385-300-0 Vol. 8 ISBN 0-88385-309-4

Printed in the United States ofAmerica

Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathe matical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City University of New York. In making this gift, Professor Dolciani, herself an exceptionally talented and successful expositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. The Association, for its part, was delighted to accept the gracious gesture initiating the revolving fund for this series from one who has served the Asso ciation with distinction, both as a member of the Committee on Publications and as a member of the Board of Governors. It was with genuine pleasure that the Board chose to name the series in her honor. The books in the series are selected for their lucid expository style and stimulating mathematical content. Typically, they contain an ample supply of exercises, many with accompanying solutions. They are intended to be sufficiently elementary for the undergraduate and even the mathematically inclined high-school student to understand and enjoy, but also to be in teresting and sometimes challenging to the more advanced mathematician.

The following DOLCIANI MATHEMATICAL EXPOSITIONS have been published.

Volume 1: MATHEMATICAL GEMS, by Ross Honsberger

Volume 2: MATHEMATICAL GEMS II, by Ross Honsberger

Volume 3: MATHEMATICAL MORSELS, by Ross Honsberger

Volume 4: MATHEMATICAL PLUMS, edited by Ross Honsberger

Volume 5: GREAT MOMENTS IN MATHEMATICS (BEFORE 1650), by Howard Eves

Volume 6: MAXIMA AND MINIMA WITHOUT CALCULUS, by Ivan Niven

Volume 7: GREAT MOMENTS IN MATHEMATICS (AFTER 1650), by Howard Eves

Volume 8: MAP COLORING, POLYHEDRA, AND THE FOUR-COLOR PROBLEM, by David Barnette

PREFACE

In the summer of 1976 Kenneth Appel and Wolfgang Haken of the University of Illinois announced that they had solved the Four-Color Problem. Suddenly what had been known to several generations of mathematicians as the Four-Color Conjecture had become the Four Color Theorem. Since it had been a conjecture for over one hundred years that all maps are four-colorable, and since a great deal of mathematics was done in attempts to solve the Four-Color Conjecture, it will be called a conjecture rather than a theorem throughout most of this book. Although the Four-Color Theorem has now been proved, the math ematics developed during the numerous unsuccessful attempts is nevertheless of lasting value. Much of combinatorial mathematics had its beginings in work on the Four-Color Conjecture. The applica tions of this mathematics goes far beyond coloring problems. Some of the exercises in this book deal with results that were ob tained under the assumption that the Four-Color Conjecture was false. When you do these problems you are requested to forget for a tnoment that the conjecture has been proved. The exercises are "an important part of this book. They contain .nany of the important theorems and definitions. This was done .to prevent the exposition from becoming a "theorem, proof" type of presentation. You should read the problems and solutions even if you do not wish to try to solve them. Most of the problems will have more than one possible solution and I do not guarantee that the solutions that I have furnished are always the simplest. Many of the problems are easy, but be warned that there are a few innocent looking prob lems that tum out to be difficult. I would like to thank Don Albers, Henry Alder, Ralph Boas, Ross Honsberger, Joe Malkevitch and Ken Rebman for many helpful sug gestions during the writing of this book.

DAVID BARNETTE vii

CONTENTS

PAGE PREFACE. ••••••••••••••••••••••••••••••••••••••••••••••.•••••• vii CHAPTER ONE. EARLymSTORY-KEMPE'S "PROOF" •••••••••••••••••• 1 1. Statement of the problem 0...... 1 2. Early history ...... 2 3. Kempe's "proof" ...... 5

CHAPTER TWO. EULER'S EQUATION. ••••••••••••.•••••••••••••••••• 20 1. Some inequalities...... 20 2. Maps and graphs...... 23 3. Aproof of Euler's equation 27 4. Kempe's "proof" revisited...... 28 S. Polyhedra...... 31 6. Euler's equation on the torus...... 37

CHAPTER THREE. HAMILTONIAN CIRCUITS •••••••••••••••••••••••••• 52 1. Tait's conjecture...... 52 2. An application to chemistry...... 59 3. Non-Hamiltonian maps...... 63

CHAPTER FOUR. ISOMORPmSM AND DUALITY. •.••••••••••••••••••••• 78 1. Isomorphism...... 78 2. Propoerties of isomorphism .... ~ ...... 82 3. Duality...... 86 4. Duality and maps...... 92 S. The Five-Color Theorem...... 96 6. Duality for Polyhedra...... 98 CHAPTER FIVE. CONVEX POLYHEDRA •.•.••.•••..•••••••.••••..••••• 108 1. Polyhedral triples...... 109 2. Steinitz's Theorem...... 112 3. Eberhard's Theorem...... 117

ix X MAP COLORING, POLYHEDRA, AND THE FOUR-COLOR PROBLEM

CHAPTER SIX. EQUIVALENT FORMS AND SPECIAL CASES. ••••••••••••• •• 126 1. Edge coloring ...... 126 2. Vertex labeling...... 128 3. Polyhedra...... 134 4. Hadwiger'sConjecture 136 5. Arranged sums...... 137

CHAPI'ER SEVEN. REDUCTIONS. •••••••••••••••••••••••••.••.•.•• •• 145 1. Reducible configurations...... 145 2. The proof of the Four-Color Theorem...... 152

CHAPTEREIGHT.NUSCELLANEOUS ..••••••••.•••••••••.••.•••.•.••• 158 1. Countries with colonies ...... 158 2. Infinite maps...... 160 3. Map coloring on other surfaces...... 161 4. What good is it? ...... 164

INDEX•....•..•..•••••...•...•.•...•..•...... •.•...••• 167 INDEX

Appel, K., 152ff edge coloring, 126ff arranged sums, 137 Euler, L., 22 Euler characteristic, 37, 43, 44, SO, 51 Barnette, D., 62, 63 Euler's Equation, 22, 27, 41, 109 Euler's Inequality, 162 Capping, 68, 106, 111 Cauchy's Rigidity Theorem, 37 face, 25 Cayley, A., 4 Five-Color Theorem, 96 Chuard, J., 56 Four-Color Theorem, 152ff Circuit, 27 Franklin, P., 152 Colonies, 158ff Colony pairs, 158 Gardner, M., 152, 154 n-colorable, 5 graph, 23 3-Color Theorem, 134 graph isomorphism, 8S 5-Color Theorem, 96 Grinberg, 69 2-colorable maps, 139, 141 Grtinbaum, B., 117, 118 combinatorially regular, 3S Gustin, G., 163 complete graph, 24 Guthrie, Francis, 3, 4 component, 41 Guthrie, Frederick, 3, 4 3-connected maps~ 86 Guy, R., 164 connected graph, 25, 90 contracting edges, 136 Hadwiger, H., 136 country,2S Hadwiger's Conjecture, 136ff cross cap, 44 Haken, W., 152ff cyclically n-connected, 58 Ha~ilton, Sir W., 3, 55 Hamiltonian circuit, 55ff, 105, 147, 154 DeMorgan, A., 3ff handle, 44 Descartes, 22 handle body, 161ff discharging, 153, 155 Heawood, P. J., 5, 160, 161, 163 duality,86ff Heawood's Map, 12~ 14 Heffler, L., 163 Eberhard, V., 117 Eberhard's Theorem, 117ff incident, 88 edge,S, 24 infinite maps, 160 167 168 MAP COLORING, POLYHEDRA, AND THE FOUR-COLOR PROBLEM

irreducible maps, 145ff regular polyhedra, 15, 35ff isomorphic graphs, 85ff Reynolds, C. N., 152 isomorphic maps, 78ff ring size, 151 isomorphism, 80ff Ringel, G., 163ff

Kempe, A. B., 5ff simple polyhedron, 34, 120, 123, 134 Kempe Chains, 5ff, 92 simplicial polyhedron, 34, 68, 121, 122, Kempe's proof, 5ff, 28 124 Kirkman, T. P., 55 Steinitz, E., 113 Kozyrev, 69 Steinitz's Theorem, 113, 117, 123 Stempel, G. J., 152 Lederberg, J., 59 Story, W. E., 30 Leibniz,22 loop, 24 Tait, P. G., 53ff, 126ff map, 25 Tait's Conjecture, 52ff Mayer, J., 152, 164 Terry, C. M., 163 Mobius, A. F., 4 topological disc, 40 Moore, E. F., 151 torus, 37ft Morgan, D. A., 50 tree, 69 Multiple edge, 24 triangular graph, 146 triangular map, 65 ll-connected, 65ff truncating, 111, 113, 134ff non-Hamiltonian nlaps, 63ff Tutte, W. T., 56,67 Tutte triangle, 56 Okamura, H., 63 Tutte's map, 56ff, 68 Ore, 0.,152 Tutte's Theorem, 67 pair-coloring, 158, 165 unavoidable set, 153ff Pannewitz, E., 56 patching, 13, 15, 18 Petersen, J., 54 valence, 5, 25, 158 Petersen's, Graph, 54, 71 3-valent map, 18, 52 planar graph, 25 vertex, 5, 23 3-polyhedral graph, 112 vertex labelling, 128ff polyhedral map, 52 polyhedral pair, 109 Wegner, G., 62 polyhedral triple, 109 Welch, L. R., 163 po~hedron, 22, 31, 98, 108, 134ff Whitney, H., 65 projective plane, 42, 49 Whitney's Theorem, 65, 67, 73, 105 proper map, 95, 102 Wino, C. E., 152 reducible configurations, 145ff reductions, 52 Youngs., J. W. T., 163

8 Map Coloring, Polyhedra, and the Four-Color Problem BARNETTE AMS / MAA PRESS VOL 4-Color Cover DOLCIANI MATHEMATICAL EXPOSITIONS MATHEMATICAL DOLCIANI / MAA AMS