Fractional Graph Theory A Rational Approach to the Theory of Graphs Edward R. Scheinerman The Johns Hopkins University Baltimore, Maryland Daniel H. Ullman The George Washington University Washington, DC With a Foreword by Claude Berge Centre National de la Recherche Scientifique Paris, France c 2008 by Edward R. Scheinerman and Daniel H. Ullman. This book was originally published by John Wiley & Sons. The copyright has reverted to the authors. Permission is hereby granted for readers to download, print, and copy this book for free so long as this copyright notice is retained. The PDF file for this book can be found here: http://www.ams.jhu.edu/~ers/fgt To Rachel, Sandy, Danny, Jessica, Naomi, Jonah, and Bobby Contents Foreword vii Preface viii 1 General Theory: Hypergraphs 1 1.1 Hypergraph covering and packing 1 1.2 Fractional covering and packing 2 1.3 Some consequences 4 1.4 A game-theoretic approach 5 1.5 Duality and duality 6 1.6 Asymptotic covering and packing 7 1.7 Exercises 12 1.8 Notes 13 2 Fractional Matching 14 2.1 Introduction 14 2.2 Results on maximum fractional matchings 17 2.3 Fractionally Hamiltonian graphs 21 2.4 Computational complexity 26 2.5 Exercises 27 2.6 Notes 28 3 Fractional Coloring 30 3.1 Definitions 30 3.2 Homomorphisms and the Kneser graphs 31 3.3 The duality gap 34 3.4 Graph products 38 3.5 The asymptotic chromatic and clique numbers 41 3.6 The fractional chromatic number of the plane 43 3.7 The Erd˝os-Faber-Lov´asz conjecture 49 3.8 List coloring 51 3.9 Computational complexity 53 3.10 Exercises 54 3.11 Notes 56 iv Contents v 4 Fractional Edge Coloring 57 4.1 Introduction 57 4.2 An exact formula 58 4.3 The matching polytope 59 4.4 Proof and consequences 61 4.5 Computational complexity 63 4.6 Fractional total chromatic number 63 4.7 Exercises 70 4.8 Notes 71 5 Fractional Arboricity and Matroid Methods 72 5.1 Arboricity and maximum average degree 72 5.2 Matroid theoretic tools 73 5.3 Matroid partitioning 77 5.4 Arboricity again 81 5.5 Maximum average degree again 84 5.6 Duality, duality, duality, and edge toughness 85 5.7 Exercises 90 5.8 Notes 91 6 Fractional Isomorphism 93 6.1 Relaxing isomorphism 93 6.2 Linear algebra tools 96 6.3 Equitable partitions 100 6.4 Iterated degree sequences 101 6.5 The main theorem 102 6.6 Other relaxations of isomorphism 105 6.7 Exercises 108 6.8 Notes 110 7 Fractional Odds and Ends 111 7.1 Fractional topological graph theory 111 7.2 Fractional cycle double covers 113 7.3 Fractional Ramsey theory 114 7.4 Fractional domination 116 7.5 Fractional intersection number 121 7.6 Fractional dimension of a poset 125 7.7 Sperner’s theorem: a fractional perspective 127 7.8 Exercises 128 7.9 Notes 129 vi Contents A Background 131 A.1 Basic graph theory and notation 131 A.2 Hypergraphs, multigraphs, multisets, fuzzy sets 134 A.3 Linear programming 134 A.4 The subadditivity lemma 137 A.5 Exercises 138 A.6 Notes 138 Bibliography 139 Author Index 148 Subject Index 152 Foreword Graph theory is one of the branches of modern mathematics having experienced a most impressive development in recent years. In the beginning, Graph Theory was only a collection of recreational or challenging problems like Euler tours or the four coloring of a map, with no clear connection among them, or among techniques used to attach them. The aim was to get a “yes” or “no” answer to simple existence questions. Under the impulse of Game Theory, Management Sciences, and Transportation Network Theory, the main concern shifted to the maximum size of entities attached to a graph. For instance, instead of establishing the existence of a 1-factor, as did Petersen and also K¨onig (whose famous theorem on bipartite graphs was discovered 20 years earlier by Steinitz in his dissertation in Breslau), the main problem was now to study the maximum number of edges in a matching, even if not a 1-factor or “perfect matching”. In this book, Scheinerman and Ullman present the next step of this evolution: Fractional Graph Theory. Fractional matchings, for instance, belong to this new facet of an old subject, a facet full of elegant results. By developing the fractional idea, the purpose of the authors is multiple: first, to enlarge the scope of applications in Scheduling, in Operations Research, or in various kinds of assignment problems; second, to simplify. The fractional version of a theorem is frequently easier to prove than the classical one, and a bound for a “fractional” coefficient of the graph is also a bound for the classical coefficient (or suggests a conjecture). A striking example is the simple and famous theorem of Vizing about the edge-chromatic number of a graph; no similar result is known for the edge- chromatic number of a hypergraph, but the reader will find in this book an analogous statement for the “fractional” edge-chromatic number which is a theorem. The conjecture of Vizing and Behzad about the total chromatic number becomes in its fractional version an elegant theorem. This book will draw the attention of the combinatorialists to a wealth of new problems and conjectures. The presentation is made accessible to students, who could find a clear exposition of the background material and a stimulating collection of exercises. And, above all, the pleasure afforded by these pages will contribute to making Combinatorics more appealing to many. Claude Berge Paris, France vii Preface Graphs model a wide variety of natural phenomena, and it is the study of these phenomena that gives rise to many of the questions asked by pure graph theorists. For example, one motivation for the study of the chromatic number in graph theory is the well-known connection to scheduling problems. Suppose that an assortment of committees needs to be scheduled, each for a total of one hour. Certain pairs of committees, because they have a member in common, cannot meet at the same time. What is the length of the shortest time interval in which all the committees can be scheduled? Let G be the graph whose vertices are these committees, with an edge between two committees if they cannot meet at the same time. The standard answer to the scheduling question is that the length of the shortest time interval is the chromatic number of G. As an illustration, suppose that there are 5 committees, with scheduling conflicts given by the graph in Figure A. Since G can be 1 5 2 4 3 Figure A: The graph C5, colored with three colors. colored with 3 colors, the scheduling can be done in 3 hours, as is illustrated in Figure B. It is a widely held misconception that, since the chromatic number of G is 3, the schedule in Figure B cannot be improved. In fact, the 5 committees can be scheduled in two-and-a-half hours, as is illustrated in Figure C. All that is required is a willingness to allow one committee to meet for half an hour, to interrupt their meeting for a time, and later to reconvene for the remaining half hour. All that is required is a willingness to break one whole hour into fractions of an hour—to break a discrete unit into fractional parts. The minimum length of time needed to schedule committees when interruptions are permitted is not the chromatic number of G but the less well-known fractional chromatic number of G. This example illustrates the theme of this book, which is to uncover the rational side of graph theory: How can integer-valued graph theory concepts be modified so they take on nonintegral values? This “fractionalization” bug has infected other parts of mathematics. Perhaps the best- known example is the fractionalization of the factorial function to give the gamma function. Fractal geometry recognizes objects whose dimension is not a whole number [126]. And analysts consider fractional derivatives [132]. Some even think about fractional partial derivatives! We are not the first to coin the term fractional graph theory; indeed, this is not the first book viii Preface ix 12345 12:00 1:00 2:00 3:00 Figure B: A schedule for the five committees. 12345 12:00 1:00 2:00 3:00 Figure C: An improved schedule for the five committees. on this subject. In the course of writing this book we found that Claude Berge wrote a short monograph [16] on this very subject. Berge’s Fractional Graph Theory is based on his lectures delivered at the Indian Statistical Institute twenty years ago. Berge includes a treatment of the fractional matching number and the fractional edge chromatic number.1 Two decades have seen a great deal of development in the field of fractional graph theory and the time is ripe for a new overview. Rationalization We have two principal methods to convert graph concepts from integer to fractional. The first is to formulate the concepts as integer programs and then to consider the linear programming relaxation (see §A.3). The second is to make use of the subadditivity lemma (Lemma A.4.1 on page 137). It is very pleasing that these two approaches typically yield the same results. Nearly every integer-valued invariant encountered in a first course in graph theory gives rise to a fractional analogue. Most of the fractional definitions in this book can be obtained from their integer-valued coun- terparts by replacing the notion of a set with the more generous notion of a “fuzzy” set [190].