{
  "id": "1601.05762",
  "version": "v1",
  "published": "2016-01-21T19:36:22.000Z",
  "updated": "2016-01-21T19:36:22.000Z",
  "title": "Cores, joins and the Fano-flow conjectures",
  "authors": [
    "Ligang Jin",
    "Giuseppe Mazzuoccolo",
    "Eckhard Steffen"
  ],
  "comment": "10 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Fan-Raspaud Conjecture states that every bridgeless cubic graph has three 1-factors with empty intersection. A weaker one than this conjecture is that every bridgeless cubic graph has two 1-factors and one join with empty intersection. Both of these two conjectures can be related to conjectures on Fano-flows. In this paper, we show that these two conjectures are equivalent to some statements on cores and weak cores of a bridgeless cubic graph. In particular, we prove that the Fan-Raspaud Conjecture is equivalent to a conjecture proposed in [E. Steffen, 1-factor and cycle covers of cubic graphs, J. Graph Theory 78 (2015) 195-206]. Furthermore, we disprove a conjecture proposed in [G. Mazzuoccolo, New conjectures on perfect matchings in cubic graphs, Electron. Notes Discrete Math. 40 (2013) 235-238] and we propose a new version of it under a stronger connectivity assumption. The weak oddness of a cubic graph $G$ is the minimum number of odd components in the complement of a join of $G$. We obtain an upper bound of weak oddness in terms of weak cores, and thus an upper bound of oddness in terms of cores as a by-product.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-21T19:36:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C70"
    ],
    "keywords": [
      "bridgeless cubic graph",
      "fano-flow conjectures",
      "upper bound",
      "weak oddness",
      "empty intersection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160105762J"
    }
  }
}