Triple systems:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Clarendon Press
1999
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Oxford mathematical monographs
Oxford science publications |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 560 S. graph. Darst. |
| ISBN: | 0198535767 |
Internformat
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| 100 | 1 | |a Colbourn, Charles J. |d 1953- |e Verfasser |0 (DE-588)111343380 |4 aut | |
| 245 | 1 | 0 | |a Triple systems |c Charles J. Colbourn and Alexander Rosa |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Oxford |b Clarendon Press |c 1999 | |
| 300 | |a XVI, 560 S. |b graph. Darst. | ||
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Datensatz im Suchindex
| _version_ | 1804127176789852160 |
|---|---|
| adam_text | CONTENTS
0 An historical introduction 1
0.1 Beginnings 1
0.2 Kirkman s On a problem in combinations 2
0.3 Kirkman s On the puzzle of the fifteen young ladies 6
0.4 Netto, Moore, and Heffter 8
0.5 Cole, Cummings, and White 10
0.6 Fisher, Yates, and Bose 11
1 Design theoretic fundamentals 13
1.1 Designs 13
1.1.1 Balanced incomplete block designs 13
1.1.2 Pairwise balanced designs 14
1.1.3 Group divisible designs 14
1.1.4 Transversal designs and orthogonal arrays 14
1.2 Latin squares and quasigroups 15
1.3 Factorizations and graph decompositions 17
1.3.1 Edge colourings 18
1.3.2 1 factorizations 19
1.3.3 Regular factors 20
1.3.4 2 factorizations 21
1.3.5 Absence of factors 21
1.3.6 Room squares 21
2 Existence: direct methods 23
2.1 Constructions from latin squares 24
2.1.1 Equivalence with Steiner quasigroups 24
2.1.2 Bose s construction and variants 25
2.1.3 Idempotent latin squares 27
2.1.4 A complete existence proof for triple systems 27
2.2 Primes and prime powers 28
2.3 Projective and affine spaces 29
2.4 Peltesohn s construction 30
2.5 Constructions from integer sequences 32
2.6 The Schreiber Wilson construction 34
2.7 Computational methods 35
2.7.1 Backtracking 36
2.7.2 Hillclimbing 37
3 Existence: recursive methods 39
3.1 Direct and indirect products 39
x CONTENTS
3.2 Doubling constructions 40
3.3 The 2v 2 and 3v 4 constructions 43
3.4 PBDs, GDDs, and PBD closure 44
3.5 Fundamental construction of Wilson 46
3.5.1 An application to GDDs 46
4 Isomorphism and invariants 49
4.1 The computational complexity of isomorphism 49
4.2 Isomorphism algorithms for fixed index 52
4.2.1 Steiner triple systems: Miller s algorithm 52
4.2.2 Automorphisms 55
4.2.3 Fixed index: Babai Luks algorithm 55
4.3 Isomorphism invariants 57
4.3.1 Trains and compact trains 57
4.3.2 Fragments and cycle structure 58
4.3.3 Neighbourhood graphs 59
4.3.4 Intersections and clique analysis 59
4.4 Work points 60
5 Enumeration 61
5.1 Small orders 61
5.2 Asymptotics 70
5.3 Computational lower bounds 72
5.4 Work points 74
6 Subsystems and holes 77
6.1 One subsystem or hole 77
6.2 Group divisible designs 82
6.3 Simple triple systems 84
6.4 Subsystems of simple systems 85
6.5 Relatives of 3 GDDs 87
6.5.1 Modified 3 GDDs 87
6.5.2 Uniform holey 3 GDDs 88
6.5.3 Holey 3 GDDs 89
6.5.4 Incomplete 3 GDDs 89
6.6 3 GDDs with first and second associates 91
6.6.1 Two groups 93
6.6.2 Group size two 94
6.6.3 The general problem 96
6.7 GDDs and constant weight codes 97
6.8 Dimension 98
6.9 Work points 100
7 Automorphisms I: small groups 101
7.1 Orbits and orbit representatives 101
7.2 Cyclic automorphisms 101 !
CONTENTS xi
7.2.1 Direct constructions 102
7.2.2 Recursive constructions 106
7.2.3 Cyclic systems with cyclic subsystems 107
7.2.4 Multipliers and multiplier automorphisms 109
7.2.5 Triple systems with multiplier automorphisms 110
7.2.6 Enumerative results 114
7.3 Rotational automorphisms 115
7.3.1 Reverse triple systems 115
7.3.2 1 rotational systems 117
7.3.3 k rotational systems 121
7.3.4 Existence for composite orders 123
7.3.5 Recursive constructions for rotational STSs 126
7.3.6 Existence for large prime orders 129
7.4 Which permutations are automorphisms? 134
7.4.1 Fixed points 134
7.4.2 Involutions 135
7.4.3 3 cycles 136
7.4.4 Bicyclic automorphisms 136
7.4.5 Transrotational automorphisms 137
7.4.6 Other automorphisms 137
7.4.7 An application: halving triple systems 137
7.5 Automorphism free triple systems 138
7.6 Work points 140
8 Automorphisms II: large groups 141
8.1 Group actions 141
8.2 Implications from group theory 142
8.3 2 transitive STSs 144
8.4 2 homogeneous STSs 144
8.5 Block transitive STSs 145
8.6 Veblen points and quadrilaterals 147
8.7 Hall triple systems 148
8.8 Abstract groups 151
8.9 Graphical triple systems 153
8.10 Work points 154
9 Leaves and partial triple systems 155
9.1 Necessary conditions 155
9.2 Recognizing A leaves 157
9.3 Maximal partial triple systems 159
9.3.1 Maximum partial triple systems 159
9.3.2 Minimum maximal partial triple systems 161
9.3.3 The spectrum of maximal partial triple systems 163
9.4 Equitable partial triple systems 164
9.5 Quadratic leaves 165
xii CONTENTS
9.6 Subgraphs of leaves 166
9.7 An application to group testing 166
9.7.1 The structure of pairs in an HS family 168
9.7.2 Maximum WR families 169
9.7.3 Testing each item at most three times 173
9.8 Rodl s nibble method 174
9.9 Work points 176
10 Excesses and coverings 177
10.1 Necessary conditions 177
10.2 Minimum excesses 178
10.3 The spectrum for minimal coverings 180
10.4 Quadratic excesses 182
10.5 Excesses and leaves: nuclear designs 183
10.6 Work points 184
11 Embedding and its variants 185
11.1 Embedding 185
11.1.1 Finite embedding theorems 185
11.1.2 A lower bound 187
11.1.3 Linear embeddings for index one 189
11.1.4 Linear embeddings for all indices 189
11.1.5 Simple embeddings 193
11.2 Enclosing 194
11.3 Immersion and intricacy 196
11.4 Work points 197
12 Neighbourhoods 199
12.1 Graphs as neighbourhoods 199
12.2 Index two 200
12.3 Index three 202
12.4 Simple neighbourhoods 203
12.5 Neighbourhood uniform triple systems 205
12.6 Double neighbourhoods for index one 206
12.7 Uniform and perfect triple systems 206
12.8 Work points 208
13 Configurations 209
13.1 Constant and variable configurations 209
13.2 Avoidance: Pasch configurations 213
13.2.1 Stinson and Wei s construction 217
13.2.2 Bose type constructions 218
13.2.3 Lu s construction 220
13.2.4 GDD constructions 221
13.2.5 The current status 222
13.2.6 Erasure codes 224
CONTENTS xiii
13.3 Avoidance of mitres 225
13.3.1 5 sparse STSs 226
13.4 Avoidance: weakly union free systems 227
13.4.1 Direct constructions 228
13.4.2 Recursive constructions 232
13.4.3 Group testing revisited 234
13.5 Avoidance in general 236
13.6 Block intersection graphs 238
13.7 Decompositions 240
13.7.1 Four line configurations 242
13.7.2 Simultaneous decompositions 243
13.7.3 Ubiquity 245
13.8 Work points 246
14 Intersections 247
14.1 Making two STSs disjoint 247
14.2 Intersections of STSs 248
14.3 Intersections of 3 GDDs 250
14.4 Orthogonal Steiner triple systems 251
14.4.1 Constructions and uses of OGDDs 253
14.4.2 OSTSs with v = 1 (mod 6) 257
14.4.3 OSTSs with v = 3 (mod 6) 259
14.5 More OGDDs 261
14.6 Intersections of resolvable STSs 265
14.7 Work points 266
15 Large sets and partitions 267
15.1 Large sets: even index 267
15.2 Large sets: index three 269
15.3 Large sets: index one 269
15.4 Large sets of 3 GDDs 274
15.4.1 Threshold schemes 275
15.5 Mutually almost disjoint systems 276
15.6 Large sets of Kirkman triple systems 277
15.7 Overlarge sets 278
15.8 Work points 278
16 Support sizes 279
16.1 Trades: the linear algebra of designs 279
16.2 Support sizes: necessity 283
16.3 Support sizes: sufficiency 286
16.3.1 Small orders 286
16.3.2 Intersections and support 287
16.3.3 Recursive constructions with fixed index 288
16.3.4 Partitions with subsystems 291
xiv CONTENTS
16.3.5 Putting the pieces together 300
16.4 Fine structure 301
16.5 Defining sets 302
16.6 Work points 303
17 Independent sets 304
17.1 The independence number 304
17.2 Complete arcs, spanning and scattering sets 305
17.3 Partitions into complete arcs 315
17.4 Generating sets 316
17.5 Covering sets 319
17.6 An application: point codes 320
17.7 Work points 324
18 Chromatic number 325
18.1 Colourings 325
18.2 Equitable weak colourings 330
18.3 Equitable colourings with many colours 332
18.4 Balanced colourings 338
18.5 Strong colouring 339
18.6 Strict colourings and the upper chromatic number 340
18.7 Achromatic number 341
18.8 The complexity of strong and weak colouring 342
18.9 Work points 343
19 Chromatic index and resolvability 344
19.1 Partial parallel classes 344
19.2 Resolvable triple systems 345
19.3 Kirkman frames and resolvable 3 GDDs 348
19.3.1 Frames 348
19.3.2 Reverse frames and an application 350
19.3.3 Nearly Kirkman triple systems 351
19.3.4 Resolvable 3 GDDs 352
19.3.5 Kirkman school project designs 354
19.3.6 Semiframes 356
19.3.7 Resolvable coverings 357
19.3.8 {2,3} PBDs 357
19.4 Colouring triples and the chromatic index 358
19.4.1 Hanani triple systems 358
19.4.2 Orthogonal double covers 359
19.4.3 Rosa triple systems 359
19.4.4 Minimum chromatic index 365
19.4.5 Upper bound 366
19.5 Computational complexity 367
19.5.1 The complexity of block colouring 367 ;
CONTENTS xv
•
19.5.2 Generating Kirkman triple systems quickly 367
19.6 Enumeration of Kirkman triple systems 370
19.7 Kirkman triple systems with subsystems 370
19.8 Work points 372
20 Orthogonal resolutions 373
20.1 Introduction 373
20.2 Generalized Room squares of degree 3 374
20.3 Kirkman squares 375
20.4 Kirkman cubes 377
20.5 Doubly resolvable twofold triple systems 378
20.6 Room rectangles and parallelepipeds 378
20.7 Work points 380
21 STSs with two subsystems 381
21.1 Necessary conditions for O ISTSs 382
21.2 Meeting the bound 383
21.3 Near the minimum 385
21.3.1 Lattice ISTSs 386
21.3.2 Intersecting cases 387
21.3.3 Disjoint cases 389
21.3.4 Eframes 390
21.3.5 Incomplete group divisible designs 394
21.4 Consequences 396
21.5 Work points 398
22 Nested and derived triple systems 399
22.1 Nested triple systems 399
22.1.1 Index one 400
22.1.2 Higher indices 403
22.1.3 Nesting GDDs and partial triple systems 403
22.1.4 Nesting into larger blocks 404
22.2 Compatible Steiner triple systems 404
22.3 Perpendicular arrays of triple systems 406
22.4 Derived triple systems 407
22.4.1 Heterogeneity and homogeneity 408
22.5 Work points 409
23 Decomposability 410
23.1 Existence 410
23.2 The complexity of decomposition 414
23.3 A finite basis theorem 416
23.4 Partitions into indecomposable systems 416
23.5 Separations of triple systems 418
23.6 Work points 419
xvi CONTENTS
•
24 Directed triple systems 422
24.1 Existence I: basic constructions 422
24.2 Existence II: directing with conflict resolution 424
24.3 Existence III: the eulerian algorithm 426
24.4 Consequences of directability 429
24.5 Automorphisms 430
24.6 The number of directed triple systems 432
24.7 Intersections and large sets 434
24.8 Work points 441
25 Mendelsohn triple systems 442
25.1 Existence: recursions and PBD constructions 442
25.2 Orienting a triple system 444
25.3 The number of Mendelsohn triple systems 445
25.4 Cyclic and rotational MTSs 447
25.5 Subsystems, intersection, and large sets 448
25.6 Embedding partial MTSs 451
25.7 Resolvable and almost resolvable MTSs 452
25.8 Hybrid triple systems 454
25.9 Work points 456
Bibliography 457
Index 551
|
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| id | DE-604.BV012530790 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:29:11Z |
| institution | BVB |
| isbn | 0198535767 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008507562 |
| oclc_num | 40460051 |
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| owner_facet | DE-20 DE-703 |
| physical | XVI, 560 S. graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Clarendon Press |
| record_format | marc |
| series2 | Oxford mathematical monographs Oxford science publications |
| spelling | Colbourn, Charles J. 1953- Verfasser (DE-588)111343380 aut Triple systems Charles J. Colbourn and Alexander Rosa 1. publ. Oxford Clarendon Press 1999 XVI, 560 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford mathematical monographs Oxford science publications Steiner, Systèmes de Triple theorie gtt Steiner systems Kombinatorik (DE-588)4031824-2 gnd rswk-swf Tripelsystem (DE-588)4761821-8 gnd rswk-swf Tripelsystem (DE-588)4761821-8 s DE-604 Kombinatorik (DE-588)4031824-2 s Rosa, Alexander Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008507562&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Colbourn, Charles J. 1953- Rosa, Alexander Triple systems Steiner, Systèmes de Triple theorie gtt Steiner systems Kombinatorik (DE-588)4031824-2 gnd Tripelsystem (DE-588)4761821-8 gnd |
| subject_GND | (DE-588)4031824-2 (DE-588)4761821-8 |
| title | Triple systems |
| title_auth | Triple systems |
| title_exact_search | Triple systems |
| title_full | Triple systems Charles J. Colbourn and Alexander Rosa |
| title_fullStr | Triple systems Charles J. Colbourn and Alexander Rosa |
| title_full_unstemmed | Triple systems Charles J. Colbourn and Alexander Rosa |
| title_short | Triple systems |
| title_sort | triple systems |
| topic | Steiner, Systèmes de Triple theorie gtt Steiner systems Kombinatorik (DE-588)4031824-2 gnd Tripelsystem (DE-588)4761821-8 gnd |
| topic_facet | Steiner, Systèmes de Triple theorie Steiner systems Kombinatorik Tripelsystem |
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