The zero-sum 3-flow conjecture for Steiner triple systems

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A Steiner triple system STS(v){\rm STS}(v) is a 22-(v,3,1)(v,3,1) design. A zero-sum nn-flow of a design D=(X,B)\mathcal{D}=(X,\mathcal{B}) is a map

f:B⟶{±1,…,±(n−1)}f:\mathcal{B}\longrightarrow \{\pm1,\ldots,\pm(n-1)\}

such that the sum of the values over all blocks incident with each point is zero. The zero-sum 3-flow conjecture. Every STS(v){\rm STS}(v) with v>7v>7 admits a zero-sum 33-flow. This conjecture concerns the existence of bounded integer flows on the incidence structures of Steiner triple systems; the source gives no evidence of resolution.

References

Primary source

Saieed Akbari, Hamid Reza Maimani, Leila Parsaei Majd and Ian M. Wanless, “Zero-sum flows for Steiner systems”, arXiv:2101.00867 (2021).

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