Overlap in consistent cycles

Štefko Miklavič, Primož Potočnik, Stephen E Wilson

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

A (directed) cycle C in a graph Γ is called consistent provided there exists an automorphism of Γ, acting as a 1-step rotation of C. A beautiful but not well-known result of J.H. Conway states that if Γ is arc-transitive and has valence d, then there are precisely d - 1 orbits of consistent cycles under the action of Aut(Γ). In this paper, we extend the definition of consistent cycles to those which admit a k-step rotation, and call them 1/k-consistent. We investigate 1/k-consistent cycles in view of their overlap. This provides a simple proof of the original Conway's theorem, as well as a generalization to orbits of 1/k-consistent cycles. A set of illuminating examples are provided.

Original languageEnglish (US)
Pages (from-to)55-71
Number of pages17
JournalJournal of Graph Theory
Volume55
Issue number1
DOIs
StatePublished - May 2007

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Overlap
Cycle
Orbit
Automorphism
Arc of a curve
Graph in graph theory
Theorem

Keywords

  • Automorphism group
  • Consistent cycle
  • Graph
  • Symmetry

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Overlap in consistent cycles. / Miklavič, Štefko; Potočnik, Primož; Wilson, Stephen E.

In: Journal of Graph Theory, Vol. 55, No. 1, 05.2007, p. 55-71.

Research output: Contribution to journalArticle

Miklavič, Š, Potočnik, P & Wilson, SE 2007, 'Overlap in consistent cycles', Journal of Graph Theory, vol. 55, no. 1, pp. 55-71. https://doi.org/10.1002/jgt.20224
Miklavič, Štefko ; Potočnik, Primož ; Wilson, Stephen E. / Overlap in consistent cycles. In: Journal of Graph Theory. 2007 ; Vol. 55, No. 1. pp. 55-71.
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