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Zero-sum flows for triple systems

Akbari, S ; Sharif University of Technology | 2017

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  1. Type of Document: Article
  2. DOI: 10.1016/j.disc.2016.08.014
  3. Publisher: Elsevier B.V , 2017
  4. Abstract:
  5. Given a 2-(v,k,λ) design, S=(X,B), a zero-sumn-flow of S is a map f:B⟶{±1,…,±(n−1)} such that for any point x∈X, the sum of f over all the blocks incident with x is zero. It has been conjectured that every Steiner triple system, STS(v), on v points (v>7) admits a zero-sum 3-flow. We show that for every pair (v,λ) for which a triple system, TS(v,λ), exists, there exists one which has a zero-sum 3-flow, except when (v,λ)∈{(3,1),(4,2),(6,2),(7,1)}. We also give a O(λ2v2) bound on n and a recursive result which shows that every STS(v) with a zero-sum 3-flow can be embedded in an STS(2v+1) with a zero-sum 3-flow if v≡3(mod4), a zero-sum 4-flow if v≡3(mod6) and with a zero-sum 5-flow if v≡1(mod4). © 2016 Elsevier B.V
  6. Keywords:
  7. Chromatic index of Steiner systems ; Steiner system ; Steiner triple system ; Zero-sum flow
  8. Source: Discrete Mathematics ; Volume 340, Issue 3 , 2017 , Pages 416-425 ; 0012365X (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/abs/pii/S0012365X16302722