Zero-sum flows for Steiner systems

Given a \(t\)-\((v, k, \lambda)\) design, \(\mathcal{D}=(X,\mathcal{B})\), a zero-sum \(n\)-flow of \(\mathcal{D}\) is a map \(f : \mathcal{B}\longrightarrow \{\pm1,\ldots, \pm(n-1)\}\) such that for any point \(x\in X\), the sum of \(f\) over all blocks incident with \(x\) is zero. For a positive i...

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Bibliographic Details
Published in:arXiv.org 2021-01
Main Authors: Akbari, Saieed, Maimani, Hamid Reza, Leila Parsaei Majd, Wanless, Ian M
Format: Article
Language:English
Online Access:Get full text
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Summary:Given a \(t\)-\((v, k, \lambda)\) design, \(\mathcal{D}=(X,\mathcal{B})\), a zero-sum \(n\)-flow of \(\mathcal{D}\) is a map \(f : \mathcal{B}\longrightarrow \{\pm1,\ldots, \pm(n-1)\}\) such that for any point \(x\in X\), 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\((u w)\) and for an STS\((2v+7)\) for \(v\equiv 1~(\mathrm{mod}~4)\), 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.
ISSN:2331-8422
DOI:10.48550/arxiv.2101.00867