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The existence of a class of Kirkman squares of index 2

Part of: Designs and configurations

Published online by Cambridge University Press:  09 April 2009

E. R. Lamken  and
S. A. Vanstone
E. R. Lamken
Affiliation:
Department of Combinatorics and Optimization University of WaterlooWaterloo, Ontario N21 3G1, Canada
S. A. Vanstone
Affiliation:
Department of Combinatorics and Optimization University of WaterlooWaterloo, Ontario N21 3G1, Canada
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Abstract

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A Kirkman square with index λ, latinicity μ, block size k and ν points, KSk(v; μ, λ), is a t × t array (t = λ(ν−1)/μ(k − 1)) defined on a ν-set V such that (1) each point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the nonempty cells of the array is a (ν, k, λ)-BIBD. For μ = 1, the existence of a KSk(ν; μ, λ) is equivalent to the existence of a doubly resolvable (ν, k, λ)-BIBD. In this case the only complete results are for k = 2. The case k = 3, λ = 1 appears to be quite difficult although some existence results are available. For k = 3, λ = 2 the problem seems to be more tractable. In this paper we prove the existence of a KS3(ν; 1, 2) for all ν ≡ 3 (mod 12).

MSC classification

Secondary: 05B30: Other designs, configurations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 1 , February 1988 , pp. 33 - 41
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Colbourn, C. J. and Vanstone, S. A., ‘Doubly resolvable twofold triple systems’, Congress. Numer. 34 (1982), 219–223.Google Scholar
[2]Colbourn, C. J., Manson, K. E. and Wallis, W. D., ‘Frames for twofold triple systems’, Ars Combin. 17 (1984), 69–78.Google Scholar
[3]Fuji-Hara, R. and Vanstone, S. A., ‘On the spectrum of doubly resolvable designs’, Congress Numer. 28 (1980), 399–407.Google Scholar
[4]Gibbons, P. and Mathon, R., ‘Construction methods for Bhaskar Rao and related designs’, J. Austral. Math. Soc. (to appear).Google Scholar
[5]Lamken, E. R., Coverings, orthogonally resolvable designs and related combinatorial configurations (Ph.D. Thesis, Univ. of Michigan, 1983).Google Scholar
[6]Morgan, E. J., ‘Some small quasi-multiple designs’, Ars Combin. 3 (1977), 233–250.Google Scholar
[7]Phelps, K. T. and Rosa, A., ‘Steiner triple systems with rotational automorphisms’, Discrete Math. 33 (1981), 57–66.CrossRefGoogle Scholar
[8]Rosa, A. and Vanstone, S. A., ‘Starter-adder techniques for Kirkman squares and Kirkman cubes of small sides’, Ars Combin. 14 (1982), 199–212.Google Scholar
[9]Rosa, A. and Vanstone, S. A., ‘On the existence of strong Kirkman cubes of order 39 and block size 3’, Ann. Discrete Math. 26 (1983), 309–320.Google Scholar
[10]Skolem, M., ‘On certain distributions of integers in pairs with given differences’, Math. Scand. 5 (1957), 57–68.CrossRefGoogle Scholar
[11]Stinson, D. R. and Vanstone, S. A., ‘A Kirkman square of order 51 and block size 3’, Discrete Math. 55 (1985), 107–111.CrossRefGoogle Scholar
[12]Stinson, D. R. and Vanstone, S. A., ‘Orthogonal packings in PG(5, 2)’, A equationes Math. 31 (1986), 159–168.CrossRefGoogle Scholar
[13]Vanstone, S. A., ‘On mutually orthogonal resolutions and near resolutions’, Ann. Discrete Math. 15 (1982), 357–369.Google Scholar
[14]Vanstone, S. A., ‘Doubly resolvable designs’, Discrete Math. 29 (1980), 77–86.CrossRefGoogle Scholar