Handcuffed designs |
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Authors: | Stephen H Y Hung N S Mendelsohn |
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Institution: | 1. University of Manitoba, Winnipeg, Manitoba, Canada
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Abstract: | Handcuffed designs are a particular case of block designs on graphs. A handcuffed design with parametersv, k, λ consists of a system of orderedk-subsets of av-set, called handcuffed blocks. In a block {A 1,A 2,?, A k } each element is assumed to be handcuffed to its neighbours and the block containsk ? 1 handcuffed pairs (A 1,A 2), (A 2,A 3), ? (A k?1,A k ). These pairs are considered unordered. The collection of handcuffed blocks constitute a hundcuffed design if the following are satisfied: (1) each element of thev-set appears amongst the blocks the same number of times (and at most once in a block) and (2) each pair of distinct elements of thev-set are handcuffed in exactly λ of the blocks. If the total number of blocks isb and each element appears inr blocks the following conditions are necessary for the handcuffed design to exist: - λv(v?1) = (k?1) b,
- rv = kb.
We denote byH(v, k, λ) the class of all handcuffed designs with parametersv, k, λ and sayH (v, k, λ) exists if there is a design with parametersv, k, λ. In this paper we prove that the necessary conditions forH (v, k, λ) exist are also sufficient in the following cases: (a)λ = 1 or 2; (b)k = 3; (c)k is evenk = 2h, and (λ, 2h ? 1) = 1; (d)k is odd,k = 2h + 1, and (λ, 4h)=2 or (λ, 4h)=1. |
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