## Combinatorial Design TheoryCombinatorial design theory is a vibrant area of combinatorics, connecting graph theory, number theory, geometry, and algebra with applications in experimental design, coding theory, and numerous applications in computer science. This volume is a collection of forty-one state-of-the-art research articles spanning all of combinatorial design theory. The articles develop new methods for the construction and analysis of designs and related combinatorial configurations; both new theoretical methods, and new computational tools and results, are presented. In particular, they extend the current state of knowledge on Steiner systems, Latin squares, one-factorizations, block designs, graph designs, packings and coverings, and develop recursive and direct constructions. The contributions form an overview of the current diversity of themes in design theory for those peripherally interested, while researchers in the field will find it to be a major collection of research advances. The volume is dedicated to Alex Rosa, who has played a major role in fostering and developing combinatorial design theory. |

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### Contents

1 | |

27 | |

Chapter 3 Pairwise Balanced Designs with Prime Power Block Sizes Exceeding 7 | 43 |

Chapter 4 Conjugate Orthogonal Latin Squares with EqualSized Holes | 65 |

Chapter 5 On Regular Packings and Coverings | 81 |

Chapter 6 An Inequality on the Parameters of Distance Regular Graphs and the Uniqueness of a Graph Related to M23 | 101 |

Chapter 7 Partitions into Indecomposable Triple Systems | 107 |

Chapter 8 Cubic Neighbourhoods in Triple Systems | 119 |

Chapter 22 A Product Theorem for Cyclic Graph Designs | 287 |

Chapter 23 A New Class of Symmetric Divisible Designs | 297 |

Chapter 24 225106 Designs Invariant under the Dihedral Group of Order Ten | 301 |

Chapter 25 On the Steiner Systems S2425 Invariant under a Group of Order 9 | 307 |

Chapter 26 Simple 5286λ Designs from PSL 227 | 315 |

Chapter 27 The Existence of Partitioned Balanced Tournament Designs of Side 4n+3 | 319 |

Chapter 28 The Existence of Partitioned Balanced Tournament Designs | 339 |

Chapter 29 Constructions for Cyclic Steiner 2Designs | 353 |

Chapter 9 The Geometry of Subspaces of an Sλ23v | 137 |

Chapter 10 On 3Blocking Sets in Projective Planes | 145 |

Chapter 11 Star SubRamsey Numbers | 153 |

Chapter 12 Colored Packing of Sets | 165 |

Chapter 13 Balanced Room Squares from Finite Geometries and their Generalizations | 179 |

Chapter 14 On the Number of Pairwise Disjoint Blocks in a Steiner System | 189 |

Chapter 15 On Steiner Systems S3526 | 197 |

Chapter 16 Halving the Complete Design | 207 |

Chapter 17 Outlines of Latin Squares | 225 |

Chapter 18 The Flower Intersection Problem for Steiner Triple Systems | 243 |

Chapter 19 Embedding Totally Symmetric Quasigroups | 249 |

Chapter 20 Cyclic Perfect One Factorizations of K2n | 259 |

Chapter 21 On Edge but not Vertex Transitive Regular Graphs | 273 |

Chapter 30 On the Spectrum of Imbrical Designs | 363 |

Chapter 31 Some Remarks on nClusters on Cubic Curves | 371 |

Chapter 32 A Few More BIBDs with k 6 and λ 1 | 379 |

Chapter 33 Isomorphism Problems for Cyclic Block Designs | 385 |

Chapter 34 Multiply Perfect Systems of Difference Sets | 393 |

Chapter 35 Some Remarks on Focal Graphs | 409 |

Chapter 36 Some Perfect OneFactorizations of K14 | 419 |

Chapter 37 A Construction for Orthogonal Designs with Three Variables | 437 |

Chapter 38 Ismorphism Classes of Small Covering Designs with Block Size Five | 441 |

Chapter 39 Graphs which are not Leaves of Maximal Partial Triple Systems | 449 |

Chapter 40 Symmetric 2 31103 Designs with Automorphisms of Order Seven | 461 |

Chapter 41 Embeddings of Steiner Systems S24v | 465 |

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### Common terms and phrases

1-factor 3-blocking set algorithm apply Lemma array automorphism group BIBD block designs block size cell classes of K3’s Colbourn coloured cl column Combinatorial complete graph construction contains Corollary cubic curve cubic graph cycle cyclic perfect decomposed decomposition deﬁned denote difference sets Discrete Math disjoint EB(P edge-colouring edges of G elements Elsevier Science Publishers exactly exists ﬁrst focal graph free loop graph G Graph Theory group-type hamiltonian cycle Hence indecomposable integers isomorphic Lemma Let G matrix modulo multigraph obtain one-factors orbits orthogonal Latin squares p,q,r)-latin rectangle P1Fs pair of orthogonal parallel classes parameters path of length PBTD(n perfect one factorizations Petersen graph positive integer prime power problem Proof prove result Room squares Rosa Science Publishers B.V. semisymmetric graphs sequencing skew resolution Steiner system Steiner triple systems subgraph Suppose symbol symmetric Theorem Theory uncoloured edge vertices

### References to this book

Block Designs: A Randomization Approach: Volume II: Design Tadeusz Calinski,Sanpei Kageyama No preview available - 2002 |