Class-uniformly resolvable pairwise balanced designs with block size two and three.

*(English)*Zbl 0749.05011A pairwise balanced design (PBD) is a pair \((X,B)\), where \(X\) is a set of points and \(B\) is a collection of subsets of \(X\) called blocks, such that each pair of points is contained in exactly one block. A parallel class of blocks in a PBD is a subset of \(B\) which partitions the point set, and a PBD is called resolveable if \(B\) admits a partition \(B_ 1,\ldots,B_ k\) into parallel classes. A class-uniformly resolvable pairwise balanced design \(\text{CURD}(K;p,r)\) is a PBD on \(p\) points, with block sizes from the set \(K\), whose block set can be resolved into \(r\) parallel classes, each parallel class containing a fixed number \(a_ k\) of blocks of size \(k\in K\). The authors indicate why such designs arise, and give some examples for \(K=\{2,3\}\).

Reviewer: E.J.F.Primrose (Leicester)

##### MSC:

05B05 | Combinatorial aspects of block designs |

##### Keywords:

pairwise balanced designs
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\textit{E. R. Lamken} et al., Discrete Math. 92, No. 1--3, 197--209 (1991; Zbl 0749.05011)

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##### References:

[1] | A.M. Assaf and A. Hartman, Resolvable group-divisible designs with block size 3, Ann. Discrete Math., to appear. · Zbl 0714.05007 |

[2] | Baker, R.D.; Wilson, R.M., Nearly kirkman triple systems, Utilitas math., 14, 289-296, (1977) · Zbl 0362.05030 |

[3] | Beth, T.; Jungnickel, D.; Lenz, H., Design theory, (1986), Cambridge Univ. Press Cambridge |

[4] | Brouwer, A.E., Two new nearly kirkman triple systems, Utilitas math., 13, 311-314, (1978) · Zbl 0379.05008 |

[5] | Ray-Chaudhuri, D.K.; Wilson, R.M., Solution of Kirkman’s schoolgirl problem, Proc. sympos. pure math., 187-203, (1971) · Zbl 0248.05009 |

[6] | Folkman, J.; Fulkerson, D.R., Edge colorings in bipartite graphs, (), 561-577 · Zbl 0204.57002 |

[7] | Hartman, A.; Rosa, A., Cyclic one-factorizations of the complete graph, European J. combin., 6, 45-48, (1985) · Zbl 0624.05051 |

[8] | R. Rees, The existence of restricted resolvable designs I: (1, 2)-factorizations of K2n, Discrete Math., to appear. |

[9] | R. Rees, The existence of restricted resolvable designs II: (1, 2)-factorizations of K2n + 1, Discrete Math., to appear. |

[10] | Rees, R.; Stinson, D.R., On resolvable group-divisible designs with block size 3, Ars combin., 23, 107-120, (1987) · Zbl 0621.05004 |

[11] | Rees, R., Uniformly resolvable pairwise balanced designs with blocksizes two and three, J. combin. theory ser. A, 45, 207-225, (1987) · Zbl 0659.05016 |

[12] | R. Rees, Cyclic (0, 1)-factorizations of the complete graph, J. Combin. Math. Combin. Comput., to appear. · Zbl 0686.05040 |

[13] | Rees, R.; Wallis, W.D., A class of resolvable pairwise balanced designs, Congr. numer., 55, 211-220, (1986) · Zbl 0611.05008 |

[14] | R. Rees, The spectrum of restricted resolvable designs with r = 2, submitted. · Zbl 0748.05037 |

[15] | Stinson, D.R., Applications and generalizations of the variance method in combinatorial designs, Utilitas math., 22, 323-333, (1982) · Zbl 0506.05007 |

[16] | Stinson, D.R., Frames for kirkman triple systems, Discrete math., 65, 289-300, (1987) · Zbl 0651.05015 |

[17] | R.M. Wilson, Construction and uses of pairwise balanced designs, combinatorics, Math. Centre Tracts 55, 18-41. |

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