{
  "id": "2101.00867",
  "version": "v1",
  "published": "2021-01-04T10:26:49.000Z",
  "updated": "2021-01-04T10:26:49.000Z",
  "title": "Zero-sum flows for Steiner systems",
  "authors": [
    "Saieed Akbari",
    "Hamid Reza Maimani",
    "Leila Parsaei Majd",
    "Ian M. Wanless"
  ],
  "journal": "Discrete Math. 343 (2020), 112074",
  "doi": "10.1016/j.disc.2020.112074",
  "categories": [
    "math.CO"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-04T10:26:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B05",
      "05B20",
      "05C21"
    ],
    "keywords": [
      "zero-sum flows",
      "steiner systems",
      "steiner quadruple systems",
      "cyclic sts",
      "blocks incident"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}