The Spectrum of Tetrahedral Quadruple Systems |
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Authors: | Jian Wang Miao Liang Beiliang Du |
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Institution: | 1. Nantong Vocational College, Nantong, 226007, People??s Republic of China 2. Department of Foundation, Suzhou Vocational University, Suzhou, 215104, People??s Republic of China 3. Department of Mathematics, Suzhou University, Suzhou, 215006, People??s Republic of China
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Abstract: | An ordered analogue of quadruple systems is tetrahedral quadruple systems. A tetrahedral quadruple system of order v and index λ, TQS(v, λ), is a pair (S, T){(S, \mathcal{T})} where S is a finite set of v elements and T{\mathcal{T}} is a family of oriented tetrahedrons of elements of S called blocks, such that every directed 3-cycle on S is contained in exactly λ blocks of T{\mathcal{T}} . When λ = 1, the spectrum problem of TQS(v, 1) has been completely determined. It is proved that a TQS(v, λ) exists if and only if λ(v − 1)(v − 2) ≡ 0 (mod 3), λv(v − 1)(v − 2) ≡ 0 (mod 4) and v ≥ 4. |
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