{
  "id": "1901.07135",
  "version": "v1",
  "published": "2019-01-22T01:10:16.000Z",
  "updated": "2019-01-22T01:10:16.000Z",
  "title": "Regular maps of order $2$-powers",
  "authors": [
    "Dong-Dong Hou",
    "Yan-Quan Feng",
    "Young Soo Kwon"
  ],
  "comment": "16pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we consider the possible types of regular maps of order $2^n$, where the order of a regular map is the order of automorphism group of the map. For $n \\le 11$, M. Conder classified all regular maps of order $2^n$. It is easy to classify regular maps of order $2^n$ whose valency or covalency is $2$ or $2^{n-1}$. So we assume that $n \\geq 12$ and $2\\leq s,t\\leq n-2$ with $s\\leq t$ to consider regular maps of order $2^n$ with type $\\{2^s, 2^t\\}$. We show that for $s+t\\leq n$ or for $s+t>n$ with $s=t$, there exists a regular map of order $2^n$ with type $\\{2^s, 2^t\\}$, and furthermore, we classify regular maps of order $2^n$ with types $\\{2^{n-2},2^{n-2}\\}$ and $\\{2^{n-3},2^{n-3}\\}$. We conjecture that, if $s+t>n$ with $s<t$, then there is no regular map of order $2^n$ with type $\\{2^s, 2^t\\}$, and we confirm the conjecture for $t=n-2$ and $n-3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-22T01:10:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B25",
      "05C10"
    ],
    "keywords": [
      "classify regular maps",
      "automorphism group",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}