Table Of ContentFractional 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
(cid:2)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 Erdo˝s-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 usedto attach them. Theaim 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 (orsuggestsaconjecture). Astrikingexampleisthesimpleandfamoustheorem
of Vizing about the edge-chromatic number of a graph; no similar result is known for the edge-
chromatic numberof a hypergraph,butthe reader will findin this bookan 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. Supposethat 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 thatis requiredis awillingness to allow onecommittee to meetfor half an hour,tointerrupt
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
arepermittedisnotthechromaticnumberofGbutthelesswell-knownfractional 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-
knownexampleisthefractionalization ofthefactorial functiontogive thegammafunction. 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
1 2 3 4 5
12:00
1:00
2:00
3:00
Figure B: A schedule for the five committees.
1 2 3 4 5
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
1
fractional matching number and the fractional edge chromatic number. 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 firstcourse 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-
terpartsbyreplacing thenotion ofasetwiththemoregenerous notion ofa“fuzzy”set[190]. While
membership in a set is governed by a {0,1}-valued indicator function, membership in a fuzzy set is
governed by a [0,1]-valued indicator function. It is possible to devise fractional analogues to nearly
1More recently, Berge devotes a chapter of his monograph Hypergraphs: Combinatorics of Finite Sets [19] to
fractional transversals of hypergraphs,which includes an exploration of fractional matchings of graphs.
