Covering arrays on graphs: qualitative independence graphs and extremal set partition theory

0.95 MB·English

by  Karen Meagher

7
0
0
2
COVERING ARRAYS ON GRAPHS: QUALITATIVE
n
a
INDEPENDENCE GRAPHS AND EXTREMAL SET
J

9 PARTITION THEORY
1

]
O
C
.
h
t
a
m
Karen Meagher
[

February 2008
1
v
3
5
5
1
0
7
0
/ A Thesis
h
t submitted to the School of Graduate Studies and Research
a
m
in partial fulfillment of the requirements
:
v
for the degree of
i
X
Doctor of Philosophy in Mathematics1
r
a
c Copyright 2008
(cid:13)
by Karen Meagher, Ottawa, Canada
1
The Ph.D. Program is a joint program with Carleton University, administered by the Ottawa-
Carleton Institute of Mathematics and Statistics
I certify that I have read this thesis and that in my opin-
ion it is fully adequate, in scope and in quality, as a
dissertation for the degree of Doctor of Philosophy.
(Principal Adviser)
I certify that I have read this thesis and that in my opin-
ion it is fully adequate, in scope and in quality, as a
dissertation for the degree of Doctor of Philosophy.
I certify that I have read this thesis and that in my opin-
ion it is fully adequate, in scope and in quality, as a
dissertation for the degree of Doctor of Philosophy.
Approved for the University Committee on Graduate
Studies:
Dean of Graduate Studies & Research
ii
Abstract
There has been a good deal of research on covering arrays over the last 20 years.
Most of this work has focused on constructions, applications and generalizations of
covering arrays. The main focus of this thesis is a generalization of covering arrays,
covering arrays on graphs. The original motivation for this generalization was to
improve applications of covering arrays to testing systems and networks, but this
extension also gives us new ways to study covering arrays.
Two vectors v,w in Zn are qualitatively independent if for all ordered pairs (a,b)
k ∈
Z Z there is a position i in the vectors where (a,b) = (v ,w ). A covering array
k k i i
×
is an array with the property that any pair of rows are qualitatively independent. A
covering array on a graph is an array with a row for each vertex of the graph with the
property that any two rows which correspond to adjacent vertices are qualitatively
independent. A covering array on the complete graph is a covering array. A covering
array is optimal if it has the minimum number of columns among covering arrays with
the same number of rows.
The addition of a graph structure to covering arrays makes it possible to use
methods from graph theory to study these designs. In this thesis, we define a family
of graphs called the qualitative independence graphs. A graph has a covering array,
with given parameters, if and only if there is a homomorphism from the graph to
a particular qualitative independence graph. Cliques in qualitative independence
graphs relate to covering arrays and independent sets are connected to intersecting
partition systems.
It is known that the exact size of an optimal binary covering array can be de-
termined using Sperner’s Theorem and the Erd˝os-Ko-Rado Theorem. In this thesis,
we find good bounds on the size of an optimal binary covering array on a graph. In
addition, we determine both the chromatic number and a core of the binary qual-
itative independence graphs. Since the rows of general covering arrays correspond
to set partitions, we give extensions of Sperner’s Theorem and the Erd˝os-Ko-Rado
Theorem to set-partition systems. These results are part of a general framework to
study extremal partition systems.
The core of the binary qualitative independence graphs can be generalized to a
subgraph of a general qualitative independence graph called the uniform qualitative
independence graph. Cliques in the uniform qualitative independence graphs relate to
balancedcovering arrays. Using these graphs, we find bounds onthe size of a balanced
covering array. We give the spectra for several of these graphs and conjecture that
they are graphs in an association scheme.
We also give a new construction for covering arrays which yields many new upper
bounds on the size of optimal covering arrays.
iv
Acknowledgements
I would like to gratefully acknowledge some of the many people and organizations
that helped me throughout this degree and in writing this thesis.
First my two supervisors, Lucia Moura and Brett Stevens. It has been a pleasure
to work with them and I feel very lucky to have had two such excellent supervisors.
I have learned an incredible amount from Lucia. She has been unbelievably patient,
supportive and encouraging! This thesis would still be a scattered pile of rough notes
if it was not for her guidance and diligence. It was Brett who convinced me to do
this degree in the first place and started me on this research project;
The examiners Charles Colbourn, Mateja Sˇajna, Steven Wang and Alan Williams
who gavemany valuablesuggestions andraisedseveral interesting questions forfuture
work.
The results in Chapter 6 of this thesis were greatly improved by valuable discus-
sions with Chris Godsil and Mike Newman;
The National Science and Engineering Research Council who provided generous
financial support;
My husband Paul Dechene, who not only moved to Ottawa for me, took over all
the household duties, made me sushi and cookies, proof-read several chapters of this
thesis, supported me through the more difficult days, but also helped me bribe Lucia
with his delicious brownies;
My family and Paul’s family for being supportive through degree after degree;
Marni Mishna, my math sister, who read several of the chapters and has always
been an inspiration. Duff McLeod who helped edit a chapter, and hasn’t been the
same since. The other math students I studied with at the University of Ottawa
v
vi
and Carleton University, especially Megan Dewar (who made the people who got us
confused think I am a nicer person than I am), Mr. Paul Elliott-Magwood (whose
enlightened conversation made coming into theoffice fun), Geetha Patil (who gave me
the mysterious advice: “a thesis is like a bollywood movie”) and Sebastian Raaphorst
(whose gossip made coming into the office fun).
P
Karen Meagher
Ottawa, Canada, June 2005
Contents
Abstract iii
Acknowledgements v
1 Introduction 1
2 Covering Arrays and Related Designs 8
2.1 Mutually Orthogonal Latin Squares . . . . . . . . . . . . . . . . . . . 8
2.2 Transversal Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Orthogonal Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Covering Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 Constructions for Covering Arrays . . . . . . . . . . . . . . . . 18
2.4.2 Asymptotic Results . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Extremal Set Theory 30
3.1 Set Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.1 t-Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.2 Set Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.3 Hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Sperner Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Bollob´as’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 The Erd˝os-Ko-Rado Theorem . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Intersecting Set Systems . . . . . . . . . . . . . . . . . . . . . 38
3.3.2 The Erd˝os-Ko-Rado Theorem . . . . . . . . . . . . . . . . . . 39
vii
CONTENTS viii
3.3.3 Generalizations of the Erd˝os-Ko-Rado Theorem . . . . . . . . 39
3.4 Application of the Erd˝os-Ko-Rado Theorem . . . . . . . . . . . . . . 41
3.4.1 Qualitatively Independent Subsets . . . . . . . . . . . . . . . . 41
4 Graph Theory 43
4.1 Basic Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1.1 Graph Homomorphism . . . . . . . . . . . . . . . . . . . . . . 44
4.1.2 Colourings, Cliques and Independent Sets . . . . . . . . . . . 44
4.1.3 Vertex-Transitive Graphs . . . . . . . . . . . . . . . . . . . . . 45
4.1.4 Cores of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1.5 Kneser Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Algebraic Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.1 Spectral Theory of Graphs . . . . . . . . . . . . . . . . . . . . 50
4.2.2 Equitable partitions . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.3 Association Schemes . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.4 Ratio Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5 Covering Arrays on Graphs 58
5.1 Bounds from Homomorphisms . . . . . . . . . . . . . . . . . . . . . . 60
5.2 Qualitative Independence Graphs . . . . . . . . . . . . . . . . . . . . 61
5.2.1 Qualitatively Independent Partitions . . . . . . . . . . . . . . 61
5.2.2 Definition of Qualitative Independence Graphs . . . . . . . . . 62
5.2.3 Homomorphisms and Qualitative Independence Graphs . . . . 65
5.3 Binary Qualitative Independence Graphs . . . . . . . . . . . . . . . . 67
5.3.1 Formulae for ω(QI(n,2)) and χ(QI(n,2)) . . . . . . . . . . . 68
5.3.2 Cores of the Binary Qualitative Independence Graphs . . . . . 72
5.4 The graphs QI(k2,k) . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6 Uniform Qualitative Independence Graphs 78
6.1 Uniform Qualitative Independence Graphs . . . . . . . . . . . . . . . 79
6.2 Bounds from Vertex Transitivity . . . . . . . . . . . . . . . . . . . . . 80
6.3 Eigenvalues of Qualitative Independence Graphs . . . . . . . . . . . . 85
CONTENTS ix
6.3.1 Eigenvalues for UQI(n,2) and AUQI(n,2) . . . . . . . . . . . 85
6.3.2 Eigenvalues for QI(k2,k) . . . . . . . . . . . . . . . . . . . . . 86
6.4 Equitable Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.4.1 Eigenvalues for QI(9,3) . . . . . . . . . . . . . . . . . . . . . 91
6.4.2 A Simple Equitable Partition . . . . . . . . . . . . . . . . . . 94
6.4.3 A Better Equitable Partition . . . . . . . . . . . . . . . . . . . 96
6.4.4 Multiplicities of the Eigenvalues . . . . . . . . . . . . . . . . . 99
6.4.5 Eigenvalues for Several Small Uniform Qualitative Indepen-
dence Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.5 Other Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.5.1 Partitions in 12 . . . . . . . . . . . . . . . . . . . . . . . . . 102
U3
6.5.2 Partitions in 15 . . . . . . . . . . . . . . . . . . . . . . . . . 104
U3
7 Partition Systems 106
7.1 Sperner Partition Systems . . . . . . . . . . . . . . . . . . . . . . . . 107
7.1.1 Sperner’s Theorem for Partition Systems in ck . . . . . . . . 108
Pk
7.1.2 A Bound on the Cardinality of Sperner Partition Systems in n 110
Pk
7.1.3 Asymptotic Growth of Maximal Sperner Partition Systems . . 112
7.2 Intersecting Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2.1 Erd˝os-Ko-Rado Theorem for Partitions for c = 2 . . . . . . . . 117
6
7.2.2 General Erd˝os-Ko-Rado Theorem for Partitions . . . . . . . . 119
7.3 Intersecting Packing Systems . . . . . . . . . . . . . . . . . . . . . . . 125
7.4 Complete Theorem for Intersecting Partition Systems . . . . . . . . . 127
7.5 Partially Intersecting Partitions . . . . . . . . . . . . . . . . . . . . . 129
8 Higher Order Problems 135
8.1 Types of Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.2 Uniform Partition Systems . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2.1 Uniform Partition System Graphs . . . . . . . . . . . . . . . . 138
8.2.2 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . 139
8.2.3 Incomparable and Comparable . . . . . . . . . . . . . . . . . . 140
8.2.4 Disjoint and Intersecting . . . . . . . . . . . . . . . . . . . . . 142
CONTENTS x
8.3 Non-Uniform Partitions . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.3.1 Incomparable . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.3.2 Comparable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.3.3 Disjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.3.4 Intersecting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
9 Conclusion 147
9.1 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.2 Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
A Tables of Bounds for CAN(r,k) 154
Bibliography 154
Index 163