{
  "id": "2007.08324",
  "version": "v1",
  "published": "2020-07-16T13:31:54.000Z",
  "updated": "2020-07-16T13:31:54.000Z",
  "title": "The mod $k$ chromatic index of graphs is $O(k)$",
  "authors": [
    "Fábio Botler",
    "Lucas Colucci",
    "Yoshiharu Kohayakawa"
  ],
  "comment": "3 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\chi'_k(G)$ denote the minimum number of colors needed to color the edges of a graph $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to $1 \\pmod k$. Scott [{\\em Discrete Math. 175}, 1-3 (1997), 289--291] proved that $\\chi'_k(G)\\leq5k^2\\log k$, and thus settled a question of Pyber [{\\em Sets, graphs and numbers} (1992), pp. 583--610], who had asked whether $\\chi_k'(G)$ can be bounded solely as a function of $k$. We prove that $\\chi'_k(G)=O(k)$, answering affirmatively a question of Scott.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-16T13:31:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "chromatic index",
      "minimum number",
      "degrees congruent",
      "discrete math"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}