{
  "id": "1604.00235",
  "version": "v1",
  "published": "2016-04-01T13:30:13.000Z",
  "updated": "2016-04-01T13:30:13.000Z",
  "title": "Decomposing graphs into a constant number of locally irregular subgraphs",
  "authors": [
    "Julien Bensmail",
    "Martin Merker",
    "Carsten Thomassen"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph is locally irregular if no two adjacent vertices have the same degree. The irregular chromatic index $\\chi_{\\rm irr}'(G)$ of a graph $G$ is the smallest number of locally irregular subgraphs needed to edge-decompose $G$. Not all graphs have such a decomposition, but Baudon, Bensmail, Przyby{\\l}o, and Wo\\'zniak conjectured that if $G$ can be decomposed into locally irregular subgraphs, then $\\chi_{\\rm irr}'(G)\\leq 3$. In support of this conjecture, Przyby{\\l}o showed that $\\chi_{\\rm irr}'(G)\\leq 3$ holds whenever $G$ has minimum degree at least $10^{10}$. Here we prove that every bipartite graph $G$ which is not an odd length path satisfies $\\chi_{\\rm irr}'(G)\\leq 10$. This is the first general constant upper bound on the irregular chromatic index of bipartite graphs. Combining this result with Przyby{\\l}o's result, we show that $\\chi_{\\rm irr}'(G) \\leq 328$ for every graph $G$ which admits a decomposition into locally irregular subgraphs. Finally, we show that $\\chi_{\\rm irr}'(G)\\leq 2$ for every $16$-edge-connected bipartite graph $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-01T13:30:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "locally irregular subgraphs",
      "constant number",
      "decomposing graphs",
      "irregular chromatic index",
      "bipartite graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160400235B"
    }
  }
}