{
  "id": "1502.04096",
  "version": "v1",
  "published": "2015-02-13T19:31:51.000Z",
  "updated": "2015-02-13T19:31:51.000Z",
  "title": "Zero-sum flows for Steiner triple systems",
  "authors": [
    "S. Akbari",
    "A. C. Burgess",
    "P. Danziger",
    "E. Mendelsohn"
  ],
  "comment": "21 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a $2$-$(v,k,\\lambda)$ design, $\\cal{S}=(X,\\cal{B})$, a {\\it zero-sum $n$-flow} of $\\cal{S}$ is a map $f: \\cal{B} \\longrightarrow \\{\\pm 1, \\ldots ,\\pm (n-1)\\}$ such that for any point $x\\in X$, the sum of $f$ around 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 $(v,\\lambda)$, for which a triple system, TS$(v,\\lambda)$ exists, there exists one which has a zero-sum $3$-flow, except when $(v,\\lambda)\\in\\{(3,1), (4,2), (6,2), (7,1)\\}$ and except possibly when $v \\equiv 10\\pmod{12}$ and $\\lambda = 2$. We also give a $O(\\lambda^2v^2)$ 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 $v\\equiv 3 \\pmod 4$, a zero-sum $4$-flow if $v\\equiv 3 \\pmod 6$ and with a zero-sum $5$-flow if $v\\equiv 1 \\pmod 4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-13T19:31:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B05",
      "05B20",
      "05C15",
      "05C21"
    ],
    "keywords": [
      "steiner triple system",
      "zero-sum flows",
      "blocks incident",
      "recursive result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}