9780198535768

Triple Systems

by ;
  • ISBN13:

    9780198535768

  • ISBN10:

    0198535767

  • Format: Hardcover
  • Copyright: 1999-07-29
  • Publisher: Clarendon Pr

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

Purchase Benefits

  • Free Shipping On Orders Over $35!
    Your order must be $35 or more to qualify for free economy shipping. Bulk sales, PO's, Marketplace items, eBooks and apparel do not qualify for this offer.
  • Get Rewarded for Ordering Your Textbooks! Enroll Now
List Price: $300.00 Save up to $30.00
  • Rent Book $270.00
    Add to Cart Free Shipping

    TERM
    PRICE
    DUE

Supplemental Materials

What is included with this book?

  • The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.
  • The Rental copy of this book is not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

Summary

Triple systems are among the simplest combinatorial designs, and are a natural generalization of graphs. They have connections with geometry, algebra, group theory, finite fields, and cyclotomy; they have applications in coding theory, cryptography, computer science, and statistics. Triplesystems provide in many cases the prototype for deep results in combinatorial design theory; this design theory is permeated by problems that were first understood in the context of triple systems and then generalized. Such a rich set of connections has made the study of triple systems anextensive, but sometimes disjointed, field of combinatorics. This book attempts to survey current knowledge on the subject, to gather together common themes, and to provide an accurate portrait of the huge variety of problems and results. Representative samples of the major syles of proof techniqueare included, as is a comprehensive bibliography.

Table of Contents

An historical introduction
1(12)
Beginnings
1(1)
Kirkman's ``On a problem in combinations''
2(4)
Kirkman's ``On the puzzle of the fifteen young ladies''
6(2)
Netto, Moore, and Heffter
8(2)
Cole, Cummings, and White
10(1)
Fisher, Yates, and Bose
11(2)
Design-theoretic fundamentals
13(10)
Designs
13(2)
Balanced incomplete block designs
13(1)
Pairwise balanced designs
14(1)
Group divisible designs
14(1)
Transversal designs and orthogonal arrays
14(1)
Latin squares and quasigroups
15(2)
Factorizations and graph decompositions
17(6)
Edge-colourings
18(1)
1-factorizations
19(1)
Regular factors
20(1)
2-factorizations
21(1)
Absence of factors
21(1)
Room squares
21(2)
Existence: direct methods
23(16)
Constructions from latin squares
24(4)
Equivalence with Steiner quasigroups
24(1)
Bose's construction and variants
25(2)
Idempotent latin squares
27(1)
A complete existence proof for triple systems
27(1)
Primes and prime powers
28(1)
Projective and affine spaces
29(1)
Peltsohn's construction
30(2)
Constructions from integer sequences
32(2)
The Schreiber-Wilson construction
34(1)
Computational methods
35(4)
Backtracking
36(1)
Hillclimbing
37(2)
Existence: recursive methods
39(10)
Direct and indirect products
39(1)
Doubling constructions
40(3)
The 2v -- 2 and 3v -- 4 constructions
43(1)
PBDs, GDDs, and PBD-closure
44(2)
Fundamental construction of Wilson
46(3)
An application to GDDs
46(3)
Isomorphism and invariants
49(12)
The computational complexity of isomorphism
49(3)
Isomorphism algorithms for fixed index
52(5)
Steiner triple systems: Miller's algorithm
52(3)
Automorphisms
55(1)
Fixed index: Babai-Luks algorithm
55(2)
Isomorphism invariants
57(3)
Trains and compact trains
57(1)
Fragments and cycle structure
58(1)
Neighbourhood graphs
59(1)
Intersections and clique analysis
59(1)
Work points
60(1)
Enumeration
61(16)
Small orders
61(9)
Asymptotics
70(2)
Computational lower bounds
72(2)
Work points
74(3)
Subsystems and holes
77(24)
One subsystem or hole
77(5)
Group divisible designs
82(2)
Simple triple systems
84(1)
Subsystems of simple systems
85(2)
Relatives of 3-GDDs
87(4)
Modified 3-GDDs
87(1)
Uniform holey 3-GDDs
88(1)
Holey 3-GDDs
89(1)
Incomplete 3-GDDs
89(2)
3-GDDs with first and second associates
91(6)
Two groups
93(1)
Group size two
94(2)
The general problem
96(1)
GDDs and constant weight codes
97(1)
Dimension
98(2)
Work points
100(1)
Automorphisms I: small groups
101(40)
Orbits and orbit representatives
101(1)
Cyclic automorphisms
101(14)
Direct constructions
102(4)
Recursive constructions
106(1)
Cyclic systems with cyclic subsystems
107(2)
Multipliers and multiplier automorphisms
109(1)
Triple systems with multiplier automorphisms
110(4)
Enumerative results
114(1)
Rotational automorphisms
115(19)
Reverse triple systems
115(2)
1-rotational systems
117(4)
k-rotational systems
121(2)
Existence for composite orders
123(3)
Recursive constructions for rotational STSs
126(3)
Existence for large prime orders
129(5)
Which permutations are automorphisms?
134(4)
Fixed points
134(1)
Involutions
135(1)
3-cycles
136(1)
Bicyclic automorphisms
136(1)
Transrotational automorphisms
137(1)
Other automorphisms
137(1)
An application: halving triple systems
137(1)
Automorphism-free triple systems
138(2)
Work points
140(1)
Automorphisms II: large groups
141(14)
Group actions
141(1)
Implications from group theory
142(2)
2-transitive STSs
144(1)
2-homogeneous STSs
144(1)
Block-transitive STSs
145(2)
Veblen points and quadrilaterals
147(1)
Hall triple systems
148(3)
Abstract groups
151(2)
Graphical triple systems
153(1)
Work points
154(1)
Leaves and partial triple systems
155(22)
Necessary conditions
155(2)
Recognizing &lamda;-leaves
157(2)
Maximal partial triple systems
159(5)
Maximum partial triple systems
159(2)
Minimum maximal partial triple systems
161(2)
The spectrum of maximal partial triple systems
163(1)
Equitable partial triple systems
164(1)
Quadratic leaves
165(1)
Subgraphs of leaves
166(1)
An application to group testing
166(8)
The structure of pairs in an HS-family
68(101)
Maximum WR-families
169(4)
Testing each item at most three times
173(1)
Rodl's nibble method
174(2)
Work points
176(1)
Excesses and coverings
177(8)
Necessary conditions
177(1)
Minimum excesses
178(2)
The spectrum for minimal coverings
180(2)
Quadratic excesses
182(1)
Excesses and leaves: nuclear designs
183(1)
Work points
184(1)
Embedding and its variants
185(14)
Embedding
185(9)
Finite embedding theorems
185(2)
A lower bound
187(2)
Linear embeddings for index one
189(1)
Linear embeddings for all indices
189(4)
Simple embeddings
193(1)
Enclosing
194(2)
Immersion and intricacy
196(1)
Work points
197(2)
Neighbourhoods
199(10)
Graphs as neigbourhoods
199(1)
Index two
200(2)
Index three
202(1)
Simple neighbourhoods
203(2)
Neighbourhood uniform triple systems
205(1)
Double neighbourhoods for index one
206(1)
Uniform and perfect triple systems
206(2)
Work points
208(1)
Configurations
209(38)
Constant and variable configurations
209(4)
Avoidance: Pasch configurations
213(12)
Stinson and Wei's construction
217(1)
Bose-type constructions
218(2)
Lu's construction
220(1)
GDD constructions
221(1)
The current status
222(2)
Erasure codes
224(1)
Avoidance of mitres
225(2)
5-sparse STSs
226(1)
Avoidance: weakly union-free systems
227(9)
Direct constructions
228(4)
Recursive constructions
232(2)
Group testing revisited
234(2)
Avoidance in general
236(2)
Block intersection graphs
238(2)
Decompositions
240(6)
Four-line configurations
242(1)
Simultaneous decompositions
243(2)
Ubiquity
245(1)
Work points
246(1)
Intersections
247(20)
Making two STSs disjoint
247(1)
Intersections of STSs
248(2)
Intersections of 3-GDDs
250(1)
Orthogonal Steiner triple systems
251(10)
Constructions and uses of OGDDs
253(4)
OSTSs with v υ ≡ 1 (mod 6)
257(2)
OSTSs with v υ ≡ 3 (mod 6)
259(2)
More OGDDs
261(4)
Intersections of resolvable STSs
265(1)
Work points
266(1)
Large sets and partitions
267(12)
Large sets: even index
269(1)
Large sets: index three
269(1)
Large sets: index one
269(5)
Large sets of 3-GDDs
274(2)
Threshold schemes
275(1)
Mutually almost disjoint systems
276(1)
Large sets of Kirkman triple systems
277(1)
Overlarge sets
278(1)
Work points
278(1)
Support sizes
279(25)
Trades: the linear algebra of designs
279(4)
Support sizes: necessity
283(3)
Support sizes: sufficiency
286(15)
Small orders
286(1)
Intersections and support
287(1)
Recursive constructions wit fixed index
288(3)
Partitions with subsystems
291(9)
Putting the pieces together
300(1)
Fine structure
301(1)
Defining sets
302(1)
Work points
303(1)
Independent sets
304(21)
The independence number
304(1)
Complete arcs, spanning and scattering sets
305(10)
Partitions into complete arcs
315(1)
Generating sets
316(3)
Covering sets
319(1)
An application: point codes
320(4)
Work points
324(1)
Chromatic number
325(19)
Colourings
325(5)
Equitable weak colourings
330(2)
Equitable colourings with many colours
332(6)
Balanced colourings
338(1)
Strong colouring
339(1)
Strict colourings and the upper chromatic number
340(1)
Achromatic number
341(1)
The complexity of strong and weak colouring
342(1)
Work points
343(1)
Chromatic index and resolvability
344(29)
Partial parallel classes
344(1)
Resolvable triple systems
345(3)
Kirkman frames and resolvable 3-GDDs
348(10)
Frames
348(2)
Reverse frames and an application
350(1)
Nearly Kirkman triple systems
351(1)
Resolvable 3-GDDs
352(2)
Kirkman school project designs
354(2)
Semiframes
356(1)
Resolvable coverings
357(1)
{2, 3}-PBDs
357(1)
Colouring triples and the chromatic index
358(9)
Hanani triple systems
358(1)
Orthogonal double covers
359(1)
Rosa triple systems
359(6)
Minimum chromatic index
365(1)
Upper bound
366(1)
Computational complexity
367(3)
The complexity of block-colouring
367(1)
Generating Kirkman triple systems quickly
367(3)
Enumeration of Kirkman triple systems
370(1)
Kirkman triple systems with subsystems
370(2)
Work points
372(1)
Orthogonal resolutions
373(8)
Introduction
373(1)
Generalized Room squares of degree 3
374(1)
Kirkman squares
375(2)
Kirkman cubes
377(1)
Doubly resolvable twofold triple systems
378(1)
Room rectangles and parallelepipeds
378(2)
Work points
380(1)
STSs with two subsystems
381(18)
Necessary conditions for -ISTSs
382(1)
Metting the bound
383(2)
Near the minimum
385(11)
Lattice ISTSs
386(1)
Intersecting cases
387(2)
Disjoint cases
389(1)
Eframes
390(4)
Incomplete group divisible designs
394(2)
Consequences
396(2)
Work points
398(1)
Nested and derived triple systems
399(11)
Nested triple systems
399(5)
Index one
400(3)
Higher indices
403(1)
Nesting GDDs and partial triple systems
403(1)
Nesting into larger blocks
404(1)
Compatible Steiner triple systems
404(2)
Perpendicular arrays of triple systems
406(1)
Derived triple systems
407(2)
Heterogeneity and homogeneity
408(1)
Work points
409(1)
Decomposability
410(12)
Existence
410(4)
The complexity of decomposition
414(2)
A finite basis theorem
416(1)
Partitions into indecomposable systems
416(2)
Separations of triple systems
418(1)
Work points
419(3)
Directed triple systems
422(20)
Existence I: basic constructions
422(2)
Existence II: directing with conflict resolution
424(2)
Existence III: the eulerian algorithm
426(3)
Consequences of directability
429(1)
Automorphisms
430(2)
The number of directed triple systems
432(2)
Intersections and large sets
434(7)
Work points
441(1)
Mendelsohn triple systems
442(15)
Existence: recursions and PBD constructions
442(2)
Orienting a triple system
444(1)
The number of Mendelsohn triple systems
445(2)
Cyclic and rotational MTSs
447(1)
Subsystems, intersection, and large sets
448(3)
Embedding partial MTSs
451(1)
Resolvable and almost resolvable MTSs
452(2)
Hybrid triple systems
454(2)
Work points
456(1)
Bibliography 457(94)
Index 551

Rewards Program

Write a Review