### 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 language | English (US) |
---|---|

Pages (from-to) | 55-71 |

Number of pages | 17 |

Journal | Journal of Graph Theory |

Volume | 55 |

Issue number | 1 |

DOIs | |

State | Published - May 2007 |

### Fingerprint

### Keywords

- Automorphism group
- Consistent cycle
- Graph
- Symmetry

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of Graph Theory*,

*55*(1), 55-71. https://doi.org/10.1002/jgt.20224

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

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 55, no. 1, pp. 55-71. https://doi.org/10.1002/jgt.20224

}

TY - JOUR

T1 - Overlap in consistent cycles

AU - Miklavič, Štefko

AU - Potočnik, Primož

AU - Wilson, Stephen E

PY - 2007/5

Y1 - 2007/5

N2 - 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.

AB - 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.

KW - Automorphism group

KW - Consistent cycle

KW - Graph

KW - Symmetry

UR - http://www.scopus.com/inward/record.url?scp=34248141616&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34248141616&partnerID=8YFLogxK

U2 - 10.1002/jgt.20224

DO - 10.1002/jgt.20224

M3 - Article

AN - SCOPUS:34248141616

VL - 55

SP - 55

EP - 71

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 1

ER -