{
  "id": "1604.07282",
  "version": "v1",
  "published": "2016-04-25T14:36:32.000Z",
  "updated": "2016-04-25T14:36:32.000Z",
  "title": "A blow-up lemma for approximate decompositions",
  "authors": [
    "Jaehoon Kim",
    "Daniela Kühn",
    "Deryk Osthus",
    "Mykhaylo Tyomkyn"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to longstanding decomposition problems. For instance, our results imply the following. Let $G$ be a quasi-random $n$-vertex graph and suppose $H_1,\\dots,H_s$ are bounded degree $n$-vertex graphs with $\\sum_{i=1}^{s} e(H_i) \\leq (1-o(1)) e(G)$. Then $H_1,\\dots,H_s$ can be packed edge-disjointly into $G$. The case when $G$ is the complete graph $K_n$ implies an approximate version of the tree packing conjecture of Gy\\'arf\\'as and Lehel for bounded degree trees, and of the Oberwolfach problem. We provide a more general version of the above approximate decomposition result which can be applied to super-regular graphs and thus can be combined with Szemer\\'edi's regularity lemma. In particular our result can be viewed as an extension of the classical blow-up lemma of Koml\\'os, S\\'ark\\H{o}zy and Szemer\\'edi to the setting of approximate decompositions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-25T14:36:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vertex graph",
      "approximate decomposition result",
      "bounded degree",
      "szemeredis regularity lemma",
      "longstanding decomposition problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160407282K"
    }
  }
}