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Super-simple, Pan-orientable, and Pan-decomposable BIBDs with Block Size 4 and Related Structures

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Abstract

Let v, k, and μ be positive integers. A tournament T of order k, briefly k-tournament, is a directed graph on k vertices in which there is exactly one directed edge between any two vertices. A (v, k, λ = 2μ)-BIBD is called T-orientable if for each of its blocks B, it is possible to replace B by a copy of T on the set B so that every ordered pair of distinct points appears in exactly μ k-tournaments. A (v, k, λ = 2μ)-BIBD is called pan-orientable if it is T-orientable for every possible k-tournament T. In this paper, we continue the earlier investigations and complete the spectrum for (v, 4, λ = 2μ)-BIBDs which possess both the pan-orientable property and the pan-decomposable property first introduced by Granville et al. (Graphs Comb 5:57–61, 1989). For all μ, we are able to show that the necessary existence conditions are sufficient. When λ = 2 and v > 4, our designs are super-simple, that is they have no two blocks with more than two common points. One new corollary to this result is that there exists a (v, 4, 2)-BIBD which is both super-simple and directable for all v ≡ 1, 4 (mod 6), v > 4. Finally, we investigate the existence of pan-orientable, pan-decomposable (v, 4, λ = 2μ)-BIBDs with a pan-orientable, pan-decomposable (w, 4, λ = 2μ)-BIBD as a subdesign; here we obtain complete results for λ = 2, 4, but there remain several open cases for λ = 6 (mostly for v < 4w), and the case λ = 12 still has to be investigated.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    R. J. R. Abel

  2. Department of Mathematics, Mount Saint Vincent University, Halifax, NS, B3M 2J6, Canada

    F. E. Bennett

  3. Institut für Mathematik, Universität Rostock, 18051, Rostock, Germany

    M. Grüttmüller

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  1. R. J. R. Abel
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  2. F. E. Bennett
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  3. M. Grüttmüller
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Correspondence to R. J. R. Abel.

Additional information

This research was supported in part by NSERC Grant OGP 0005320 for F. E. Bennett.

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Abel, R.J.R., Bennett, F.E. & Grüttmüller, M. Super-simple, Pan-orientable, and Pan-decomposable BIBDs with Block Size 4 and Related Structures. Graphs and Combinatorics 25, 767–788 (2009). https://doi.org/10.1007/s00373-010-0872-6

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