{
  "id": "2111.15484",
  "version": "v3",
  "published": "2021-11-30T15:19:37.000Z",
  "updated": "2021-12-20T11:44:42.000Z",
  "title": "Semi-equivelar toroidal maps and their k-semiregular covers",
  "authors": [
    "Arnab Kundu",
    "Dipendu Maity"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:2111.13085, arXiv:2110.12375",
  "categories": [
    "math.CO"
  ],
  "abstract": "If the face\\mbox{-}cycles at all the vertices in a map are of same type then the map is called semi\\mbox{-}equivelar. In particular, it is called equivelar if the face-cycles contain same type of faces. A map is semiregular (or almost regular) if it has as few flag orbits as possible for its type. A map is $k$-regular if it is equivelar and the number of flag orbits of the map $k$ under the automorphism group. In particular, if $k =1$, its called regular. A map is $k$-semiregular if it contains more number of flags as compared to its type with the number of flags orbits $k$. Drach et al. \\cite{drach:2019} have proved that every semi-equivelar toroidal map has a finite unique minimal semiregular cover. In this article, we show the bounds of flag orbits of semi-equivelar toroidal maps, i.e., there exists $k$ for each type such that every semi-equivelar map is $\\ell$-uniform for some $\\ell \\le k$. We show that none of the Archimedean types on the torus is semiregular, i.e., for each type, there exists a map whose number of flag orbits is more than its type. We also prove that if a semi-equivelar map is $m$-semiregular then it has a finite index $t$-semiregular minimal cover for $t \\le m$. We also show the existence and classification of $n$ sheeted $k$-semiregular maps for some $k$ of semi-equivelar toroidal maps for each $n \\in \\mathbb{N}$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2021-12-20T11:44:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C20",
      "52B70",
      "51M20",
      "57M60"
    ],
    "keywords": [
      "semi-equivelar toroidal map",
      "k-semiregular covers",
      "flag orbits",
      "finite unique minimal semiregular cover",
      "semi-equivelar map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}