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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

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