{
  "id": "1809.09302",
  "version": "v1",
  "published": "2018-09-25T03:20:10.000Z",
  "updated": "2018-09-25T03:20:10.000Z",
  "title": "Partitioning The Edge Set of a Hypergraph Into Almost Regular Cycles",
  "authors": [
    "Amin Bahmanian",
    "Sadegheh Haghshenas"
  ],
  "comment": "17 pages",
  "journal": "Journal of Combinatorial Designs, Volume 26, Issue 10, October 2018, Pages 465-479",
  "doi": "10.1002/jcd.21610",
  "categories": [
    "math.CO"
  ],
  "abstract": "A cycle of length $t$ in a hypergraph is an alternating sequence $v_1,e_1,v_2\\dots,v_t,e_t$ of distinct vertices $v_i$ and distinct edges $e_i$ so that $\\{v_i,v_{i+1}\\}\\subseteq e_i$ (with $v_{t+1}:=v_1$). Let $\\lambda K_n^h$ be the $\\lambda$-fold $n$-vertex complete $h$-graph. Let $\\mathcal G=(V,E)$ be a hypergraph all of whose edges are of size at least $h$, and $2\\leq c_1\\leq \\dots\\leq c_k\\leq |V|$. In order to partition the edge set of $\\mathcal G$ into cycles of specified lengths $c_1, \\dots, c_k$, an obvious necessary condition is that $\\sum_{i=1}^k c_i=|E|$. We show that this condition is sufficient in the following cases: (i) $h\\geq \\max\\{c_k, \\lceil n/2 \\rceil+1\\}$; (ii) $\\mathcal G=\\lambda K_n^h$, $h\\geq \\lceil n/2 \\rceil+2$; (iii) $\\mathcal G=K_n^h$, $c_1= \\dots=c_k:=c$, $c|n(n-1), n\\geq 85$. In (ii), we guarantee that each cycle is almost regular. In (iii), we also solve the case where a \"small\" subset $L$ of edges of $K_n^h$ is removed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-25T03:20:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65",
      "05C45",
      "05C70"
    ],
    "keywords": [
      "edge set",
      "regular cycles",
      "hypergraph",
      "distinct vertices",
      "distinct edges"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}