{
  "id": "1412.6237",
  "version": "v1",
  "published": "2014-12-19T07:40:46.000Z",
  "updated": "2014-12-19T07:40:46.000Z",
  "title": "New Bounds for the Acyclic Chromatic Index",
  "authors": [
    "Anton Bernshteyn"
  ],
  "comment": "10 pages, 2 figures. arXiv admin note: text overlap with arXiv:1410.1591",
  "categories": [
    "math.CO"
  ],
  "abstract": "An edge coloring of a graph $G$ is called an acyclic edge coloring if it is proper (i.e. adjacent edges receive different colors) and every cycle in $G$ contains edges of at least three different colors (there are no bichromatic cycles in $G$). The least number of colors needed for an acyclic edge coloring of $G$ is called the acyclic chromatic index of $G$ and is denoted by $a'(G)$. It is conjectured by Fiam\\v{c}ik and independently by Alon et al. that $a'(G) \\leq \\Delta(G)+2$, where $\\Delta(G)$ denotes the maximum degree of $G$. However, the best known general bound is $a'(G)\\leq 4(\\Delta(G)-1)$ due to Esperet and Parreau. We apply a generalization of the Lov\\'{a}sz Local Lemma that we call the Local Action Lemma to show that if $G$ contains no copy of a given bipartite graph $H$, then $a'(G) \\leq 3\\Delta(G)+o(\\Delta(G))$. Moreover, for every $\\varepsilon>0$ there exists a constant $c$ such that if $g(G)\\geq c$, then $a'(G)\\leq(2+\\varepsilon)\\Delta(G)+o(\\Delta(G))$, where $g(G)$ denotes the girth of $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-19T07:40:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "acyclic chromatic index",
      "acyclic edge coloring",
      "local action lemma",
      "bichromatic cycles",
      "adjacent edges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.6237B"
    }
  }
}