{
  "id": "2404.13155",
  "version": "v1",
  "published": "2024-04-19T19:48:33.000Z",
  "updated": "2024-04-19T19:48:33.000Z",
  "title": "On the rectilinear crossing number of complete balanced multipartite graphs and layered graphs",
  "authors": [
    "Ruy Fabila-Monroy",
    "Rosna Paul",
    "Jenifer Viafara-Chanchi",
    "Alexandra Weinberger"
  ],
  "categories": [
    "math.CO",
    "cs.CG"
  ],
  "abstract": "A rectilinear drawing of a graph is a drawing of the graph in the plane in which the edges are drawn as straight-line segments. The rectilinear crossing number of a graph is the minimum number of pairs of edges that cross over all rectilinear drawings of the graph. Let $n \\ge r$ be positive integers. The graph $K_n^r$, is the complete $r$-partite graph on $n$ vertices, in which every set of the partition has at least $\\lfloor n/r \\rfloor$ vertices. The layered graph, $L_n^r$, is an $r$-partite graph on $n$ vertices, in which for every $1\\le i \\le r-1$, all the vertices in the $i$-th partition are adjacent to all the vertices in the $(i+1)$-th partition. In this paper, we give upper bounds on the rectilinear crossing numbers of $K_n^r$ and~$L_n^r$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-19T19:48:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rectilinear crossing number",
      "complete balanced multipartite graphs",
      "layered graph",
      "th partition",
      "straight-line segments"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}