{
  "id": "2405.11409",
  "version": "v1",
  "published": "2024-05-18T23:01:39.000Z",
  "updated": "2024-05-18T23:01:39.000Z",
  "title": "On Tuza's Conjecture in Dense Graphs",
  "authors": [
    "Luis Chahua",
    "Juan Gutierrez"
  ],
  "comment": "12 pages, 1 figure",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "In 1982, Tuza conjectured that the size $\\tau(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $\\nu(G)$ of a maximum set of edge-disjoint triangles of $G$. This conjecture was proved for several graph classes. In this paper, we present three results regarding Tuza's Conjecture for dense graphs. By using a probabilistic argument, Tuza proved its conjecture for graphs on $n$ vertices with minimum degree at least $\\frac{7n}{8}$. We extend this technique to show that Tuza's conjecture is valid for split graphs with minimum degree at least $\\frac{3n}{5}$; and that $\\tau(G) < \\frac{28}{15}\\nu(G)$ for every tripartite graph with minimum degree more than $\\frac{33n}{56}$. Finally, we show that $\\tau(G)\\leq \\frac{3}{2}\\nu(G)$ when $G$ is a complete 4-partite graph. Moreover, this bound is tight.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-18T23:01:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dense graphs",
      "minimum degree",
      "results regarding tuzas conjecture",
      "minimum set",
      "graph classes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}