{
  "id": "1302.3657",
  "version": "v2",
  "published": "2013-02-15T01:25:37.000Z",
  "updated": "2013-03-08T09:50:36.000Z",
  "title": "Some Remarks on Graphical Sequences for Graphs and Bipartite Graphs",
  "authors": [
    "Grant Cairns",
    "Stacey Mendan"
  ],
  "comment": "Paper has been rewritten for publication as two separate papers: one on graphs with loops and one on the improvement of the Zverovich--Zverovich result",
  "categories": [
    "math.CO"
  ],
  "abstract": "For finite sequence $\\underbar{\\em d}$ of positive integers, we consider graphs that have $\\underbar{\\em d}$ as their list of vertex degrees, and bipartite graphs for which each part has $\\underbar{\\em d}$ as its list of vertex degrees. In particular, we make a connection between a result for bipartite graphs by Alon, Ben-Shimon and Krivelevich and a result of Zverovich and Zverovich for graphs, and we give an improvement of a result of Zverovich and Zverovich. We show that the bipartite graphs with vertex degree sequences $(\\underbar{\\em d},\\underbar{\\em d}\\,)$ are in one to one correspondence with graphs with loops with reduced degree sequence $\\underbar{\\em d}$, where the reduced degree of a vertex is defined to be the number of edges incident to the vertex, with loops counted only once. We also give two Erd\\H{o}s--Gallai type theorems for graphs with loops.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-03-08T09:50:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C07"
    ],
    "keywords": [
      "bipartite graphs",
      "graphical sequences",
      "vertex degree sequences",
      "type theorems",
      "reduced degree sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.3657C"
    }
  }
}