{
  "id": "2305.01948",
  "version": "v1",
  "published": "2023-05-03T07:56:13.000Z",
  "updated": "2023-05-03T07:56:13.000Z",
  "title": "Upper Bounds on the Acyclic Chromatic Index of Degenerate Graphs",
  "authors": [
    "Nevil Anto",
    "Manu Basavaraju",
    "Suresh Manjanath Hegde",
    "Shashanka Kulamarva"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "An acyclic edge coloring of a graph is a proper edge coloring without any bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum $k$ such that $G$ has an acyclic edge coloring with $k$ colors. Fiam\\v{c}\\'{\\i}k conjectured that $a'(G) \\le \\Delta+2$ for any graph $G$ with maximum degree $\\Delta$. A graph $G$ is said to be $k$-degenerate if every subgraph of $G$ has a vertex of degree at most $k$. Basavaraju and Chandran proved that the conjecture is true for $2$-degenerate graphs. We prove that for a $3$-degenerate graph $G$, $a'(G) \\le \\Delta+5$, thereby bringing the upper bound closer to the conjectured bound. We also consider $k$-degenerate graphs with $k \\ge 4$ and give an upper bound for the acyclic chromatic index of the same.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-03T07:56:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "acyclic chromatic index",
      "degenerate graph",
      "acyclic edge coloring",
      "upper bound closer",
      "bichromatic cycles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}