{
  "id": "1402.3741",
  "version": "v2",
  "published": "2014-02-16T00:53:06.000Z",
  "updated": "2015-06-18T13:44:43.000Z",
  "title": "On path-cycle decompositions of triangle-free graphs",
  "authors": [
    "Andrea Jiménez",
    "Yoshiko Wakabayashi"
  ],
  "comment": "23 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this work, we study conditions for the existence of length-constrained path-cycle decompositions, that is, partitions of edge sets of a graph into paths and cycles of a given minimum length. Our main contribution is the characterization of the class of all triangle-free graphs with odd distance at least $3$ that admit a path-cycle decomposition with elements of length at least $4$. As a consequence, it follows that Gallai's conjecture on path decomposition holds in a broad class of sparse graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-16T00:53:06.000Z",
      "abstract": "Gallai conjectured that every connected graph on n vertices admits a path decomposition, i.e., a decomposition of its edge set into paths, of cardinality at most $\\lceil{n}/{2}\\rceil$. Lov\\'asz proved that such a graph has a path-cycle decomposition, i.e., a decomposition of its edge set into paths and cycles, of cardinality at most $\\lfloor{n}/{2}\\rfloor$. In this work, we study conditions for the existence of path-cycle decompositions of a graph with elements of a given minimum length. Our main contribution is the characterization of the class of all triangle-free graphs with odd distance at least 3 that do not admit a path-cycle decomposition with elements of length at least 4. We prove that this class of graphs can be recursively constructed and satisfies Gallai's conjecture. Moreover, using a result by Harding et al. we transform path-cycle decompositions with elements of length at least 4 into path decompositions with elements of average length at least 4. As a consequence, we prove that Gallai's conjecture holds in a broad subclass of planar graphs.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-18T13:44:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C38",
      "05C05",
      "05C10",
      "05C75",
      "G.2.2"
    ],
    "keywords": [
      "triangle-free graphs",
      "path decomposition",
      "edge set",
      "transform path-cycle decompositions",
      "satisfies gallais conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.3741J"
    }
  }
}