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Group Divisible Designs with Block Size Four and Type \(g^u b^1 (gu/2)^1\)

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Abstract

We discuss group divisible designs with block size four and type \(g^u b^1 (gu/2)^1\), where \(u = 5\), 6 and 7. For integers a and b, we prove the following. (i) A 4-GDD of type \((4a)^5 b^1 (10a)^1\) exists if and only if \(a \ge 1\), \(b \equiv a\) (mod 3) and \(4a \le b \le 10a\). (ii) A 4-GDD of type \((6a+3)^6 b^1 (18a+9)^1\) exists if and only if \(a \ge 0\), \(b \equiv 3\) (mod 6) and \(6a+3 \le b \le 18a + 9\). (iii) A 4-GDD of type \((6a)^6 b^1 (18a)^1\) exists if and only if \(a \ge 1\), \(b \equiv 0\) (mod 3) and \(6a \le b \le 18a\). (iv) A 4-GDD of type \((12a)^7 b^1 (42a)^1\) exists if and only if \(a \ge 1\), \(b \equiv 0\) (mod 3) and \(12a \le b \le 42a\), except possibly for \(12a \in \{120, 180, 240, 360, 420, 720, 840\}\), \(24a< b < 42a\), for \(12a \in \{144, 1008\}\), \(30a< b < 42a\), and for \(12a \in \{168, 252, 336, 504, 1512\}\), \(36a< b < 42a\).

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The author would like to thank the reviewers for suggestions from which this paper has benefitted.

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  1. School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK

    Anthony D. Forbes

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Appendix

Appendix

For the Appendix, see https://arxiv.org/abs/arXiv:1906.02170.

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Forbes, A.D. Group Divisible Designs with Block Size Four and Type \(g^u b^1 (gu/2)^1\). Graphs and Combinatorics 36, 1687–1703 (2020). https://doi.org/10.1007/s00373-020-02213-5

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  • DOI: https://doi.org/10.1007/s00373-020-02213-5

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