Zero-sum flows for Steiner systems

S. Akbari, H. R. Maimani, Leila Parsaei Majd, I. M. Wanless

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Given a t-(v,k,λ) design, D=(X,B), a zero-sum n-flow of D is a map f:B⟶{±1,…,±(n−1)} such that for any point x∈X, the sum of f over all blocks incident with x is zero. For a positive integer k, we find a zero-sum k-flow for an STS(uw) and for an STS(2v+7) for v≡1(mod4), if there are STS(u), STS(w) and STS(v) such that the STS(u) and STS(v) both have a zero-sum k-flow. In 2015, it was conjectured that for v>7 every STS(v) admits a zero-sum 3-flow. Here, it is shown that many cyclic STS(v) have a zero-sum 3-flow. Also, we investigate the existence of zero-sum flows for some Steiner quadruple systems.

Original languageEnglish
Article number112074
Number of pages8
JournalDiscrete Mathematics
Issue number11
Publication statusPublished - Nov 2020


  • Steiner quadruple system
  • Steiner triple system
  • Zero-sum flow

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