### 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) |
---|---|

Pages (from-to) | 262-273 |

Number of pages | 12 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 47 |

Issue number | 3 |

DOIs | |

State | Published - 1989 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

**Cantankerous maps and rotary embeddings of K _{n}.** / Wilson, Stephen E.

Research output: Contribution to journal › Article

_{n}',

*Journal of Combinatorial Theory. Series B*, vol. 47, no. 3, pp. 262-273. https://doi.org/10.1016/0095-8956(89)90028-2

}

TY - JOUR

T1 - Cantankerous maps and rotary embeddings of Kn

AU - Wilson, Stephen E

PY - 1989

Y1 - 1989

N2 - 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 Kn, then n must be a power of a prime. We will show that, if n > 6, Kn 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 Kn given by Heffter's construction are chiral.

AB - 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 Kn, then n must be a power of a prime. We will show that, if n > 6, Kn 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 Kn given by Heffter's construction are chiral.

UR - http://www.scopus.com/inward/record.url?scp=38249026446&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249026446&partnerID=8YFLogxK

U2 - 10.1016/0095-8956(89)90028-2

DO - 10.1016/0095-8956(89)90028-2

M3 - Article

AN - SCOPUS:38249026446

VL - 47

SP - 262

EP - 273

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 3

ER -