Handcuffed designs |
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Authors: | Stephen H.Y. Hung N.S. Mendelsohn |
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Affiliation: | National Research Council of Canada, Ottawa, Canada;University of Manitoba, Winnipeg, Canada |
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Abstract: | Handcuffed designs are a particular case of block designs on graphs. A handcuffed design with parameters v, k, λ consists of a system of ordered k-subsets of a v-set, called handcuffed blocks. In a block {A1, A2,…, Ak} each element is assumed to be handcuffed to its neighbors and the block contains k ? 1 handcuffed pairs (A1, A2), (A2, A1), …, (Ak?1, Ak). These pairs are considered unordered. The collection of handcuffed blocks constitutes a handcuffed design if the following are satisfied: (1) each element of the v-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 the v-set are handcuffed in exactly λ of the blocks. If the total number of blocks is b and each element appears in r blocks the following conditions are necessary for the handcuffed design to exist. (1) λv (v ? 1) = (k ? 1)b. (2) rv = kb. In this paper it is shown that the necessary conditions are also sufficient. |
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