Cantankerous maps and rotary embeddings of Kn

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33 Scopus citations

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 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.

Original languageEnglish (US)
Pages (from-to)262-273
Number of pages12
JournalJournal of Combinatorial Theory, Series B
Volume47
Issue number3
DOIs
StatePublished - Dec 1989

ASJC Scopus subject areas

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

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