284 Pages 63 B/W Illustrations
    by Chapman & Hall

    272 Pages 63 B/W Illustrations
    by Chapman & Hall

    Design Theory, Second Edition presents some of the most important techniques used for constructing combinatorial designs. It augments the descriptions of the constructions with many figures to help students understand and enjoy this branch of mathematics.

    This edition now offers a thorough development of the embedding of Latin squares and combinatorial designs. It also presents some pure mathematical ideas, including connections between universal algebra and graph designs.

    The authors focus on several basic designs, including Steiner triple systems, Latin squares, and finite projective and affine planes. They produce these designs using flexible constructions and then add interesting properties that may be required, such as resolvability, embeddings, and orthogonality. The authors also construct more complicated structures, such as Steiner quadruple systems.

    By providing both classical and state-of-the-art construction techniques, this book enables students to produce many other types of designs.

    Steiner Triple Systems

    The Existence Problem

    ≡ 3 (mod 6): The Bose Construction

    ≡ 1 (mod 6): The Skolem Construction

    ≡ 5 (mod 6): The 6n + 5 Construction

    Quasigroups with Holes and Steiner Triple Systems

    The Wilson Construction

    Cyclic Steiner Triple Systems

    The 2n + 1 and 2n + 7 Constructions

    λ-Fold Triple Systems

    Triple Systems of Index λ > 1

    The Existence of Indempotent Latin Squares

    2-fold Triple Systems

    λ= 3 and 6

    λ-Fold Triple Systems in General

    Quasigroup Identities and Graph Decompositions

    Quasigroup Identities

    Mendelsohn Triple Systems Revisited

    Steiner Triple Systems Revisited

    Maximum Packings and Minimum Coverings

    The General Problem

    Maximum Packings

    Minimum Coverings

    Kirkman Triple Systems

    A Recursive Construction

    Constructing Pairwise Balanced Designs

    Mutually Orthogonal Latin Squares

    Introduction

    The Euler and MacNeish Conjectures

    Disproof of the MacNeish Conjecture

    Disproof of the Euler Conjecture

    Orthogonal Latin Squares of Order ≡ 2 (mod 4)

    Affine and Projective Planes

    Affine Planes

    Projective Planes

    Connections between Affine and Projective Planes

    Connection between Affine Planes and Complete Sets of MOLS

    Coordinating the Affine Plane

    Intersections of Steiner Triple Systems

    Teirlinck’s Algorithm

    The General Intersection Problem

    Embeddings

    Embedding Latin Rectangles—Necessary Conditions

    Edge-Coloring Bipartite Graphs

    Embedding Latin Rectangles: Ryser’s Sufficient Conditions

    Embedding Idempotent Commutative Latin Squares: Cruse’s Theorem

    Embedding Partial Steiner Triple Systems

    Steiner Quadruple Systems

    Introduction

    Constructions of Steiner Quadruple Systems

    The Stern and Lenz Lemma

    The (3v – 2u)-Construction

    Appendix A: Cyclic Steiner Triple Systems

    Appendix B: Answers to Selected Exercises

    References

    Index


    Biography

    Charles C. Lindner, Christopher A. Rodger

    …it is remarkable how quickly the book propels the reader from the basics to the frontiers of design theory … Combined, these features make the book an excellent candidate for a design theory text. At the same time, even the seasoned researcher of triple systems will find this a useful resource.

    —Peter James Dukes (3-VCTR-MS; Victoria, BC), Mathematical Reviews, 2010