### Abstract

In this paper we will define a class of locally non-orientable regular maps called "cantankerous." We will show that cantankerous maps are self-Petrie, we will prove a lower bound on the number of vertices such a map may have, and we will give some data which suggest that the cantankerous maps are a fairly restricted class of regular maps. Our main result here is that any vertex-improper map must either be one of these cantankerous maps or be constructed from a smaller vertex-proper map by the Riemann-surface algorithm. We then apply these results to graph theory. Biggs has shown that if M is an orientable rotary map whose underlying graph is K_{n}, then n must be a power of a prime. We will show that, if n > 6, K_{n} has no regular embedding; this shows that the only exception to Biggs' theorem in the non-orientable case is n = 6, and that the rotary embeddings of K_{n} given by Heffter's construction are chiral.

Original language | English (US) |
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Pages (from-to) | 262-273 |

Number of pages | 12 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 47 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1989 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics