{
  "id": "1104.5113",
  "version": "v1",
  "published": "2011-04-27T11:06:19.000Z",
  "updated": "2011-04-27T11:06:19.000Z",
  "title": "An Alternative Proof of the $H$-Factor Theorem",
  "authors": [
    "Hongliang Lu",
    "Qinglin Yu"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $H: V(G) \\rightarrow 2^{\\mathbb{N}}$ be a set mapping for a graph $G$. Given a spanning subgraph $F$ of $G$, $F$ is called a {\\it general factor} or an $H$-{\\it factor} of $G$ if $d_{F}(x)\\in H(x)$ for every vertex $x\\in V(G)$. $H$-factor problems are, in general, $NP$-complete problems and imply many well-known factor problems (e.g., perfect matchings, $f$-factor problems and $(g, f)$-factor problems) as special cases. Lov\\'asz [The factorization of graphs (II), Acta Math. Hungar., 23 (1972), 223--246] gave a structure description and obtained a deficiency formula for $H$-optimal subgraphs. In this note, we use a generalized alternating path method to give a structural characterization and provide an alternative and shorter proof of Lov\\'asz's deficiency formula.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-04-27T11:06:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "factor theorem",
      "alternative proof",
      "lovaszs deficiency formula",
      "well-known factor problems",
      "structure description"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.5113L"
    }
  }
}