{
  "id": "1001.2936",
  "version": "v1",
  "published": "2010-01-18T00:31:00.000Z",
  "updated": "2010-01-18T00:31:00.000Z",
  "title": "Classification of nonorientable regular embeddings of complete bipartite graphs",
  "authors": [
    "Jin Ho Kwak",
    "Young Soo Kwon"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A 2-cell embedding of a graph $G$ into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags - mutually incident vertex-edge-face triples. In this paper, we classify the regular embeddings of complete bipartite graphs $K_{n,n}$ into nonorientable surfaces. Such regular embedding of $K_{n,n}$ exists only when $n = 2p_1^{a_1}p_2^{a_2}... p_k^{a_k}$ (a prime decomposition of $n$) and all $p_i \\equiv \\pm 1 (\\mod 8)$. In this case, the number of those regular embeddings of $K_{n,n}$ up to isomorphism is $2^k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-01-18T00:31:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C10",
      "05C30"
    ],
    "keywords": [
      "complete bipartite graphs",
      "nonorientable regular embeddings",
      "classification",
      "mutually incident vertex-edge-face triples",
      "prime decomposition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}