{
  "id": "2304.14579",
  "version": "v1",
  "published": "2023-04-28T00:46:27.000Z",
  "updated": "2023-04-28T00:46:27.000Z",
  "title": "Full Characterization of Color Degree Sequences in Complete Graphs Without Tricolored Triangles",
  "authors": [
    "Anton Trygub"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For an edge-colored complete graph, we define the color degree of a node as the number of colors appearing on edges incident to it. In this paper, we consider colorings that don't contain tricolored triangles (also called rainbow triangles); these colorings are also called Gallai colorings. We give a complete characterization of all possible color degree sequences $d_1 \\le d_2 \\le \\dots \\le d_n$ that can arise on a Gallai coloring of $K_n$: it is necessary and sufficient that \\[ \\sum_{i = k}^n \\frac{1}{2^{d_i - d_{k-1}}} \\ge 1 \\] holds for all $1 \\le k \\le n$, where $d_0=0$ for convenience. As a corollary, this gives another proof of a 2018 result of Fujita, Li, and Zhang who showed that the minimum color degree in such a coloring is at most $\\log_2n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-28T00:46:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "color degree sequences",
      "full characterization",
      "dont contain tricolored triangles",
      "minimum color degree",
      "complete characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}