{
  "id": "1705.08592",
  "version": "v1",
  "published": "2017-05-24T03:24:06.000Z",
  "updated": "2017-05-24T03:24:06.000Z",
  "title": "Sufficient conditions for the existence of a path-factor which are related to odd components",
  "authors": [
    "Yoshimi Egawa",
    "Michitaka Furuya",
    "Kenta Ozeki"
  ],
  "comment": "18 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we are concerned with sufficient conditions for the existence of a $\\{P_{2},P_{2k+1}\\}$-factor. We prove that for $k\\geq 3$, there exists $\\varepsilon_{k}>0$ such that if a graph $G$ satisfies $\\sum_{0\\leq j\\leq k-1}c_{2j+1}(G-X)\\leq \\varepsilon_{k}|X|$ for all $X\\subseteq V(G)$, then $G$ has a $\\{P_{2},P_{2k+1}\\}$-factor, where $c_{i}(G-X)$ is the number of components $C$ of $G-X$ with $|V(C)|=i$. On the other hand, we construct infinitely many graphs $G$ having no $\\{P_{2},P_{2k+1}\\}$-factor such that $\\sum_{0\\leq j\\leq k-1}c_{2j+1}(G-X)\\leq \\frac{32k+141}{72k-78}|X|$ for all $X\\subseteq V(G)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-24T03:24:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sufficient conditions",
      "odd components",
      "path-factor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}