### Abstract

Let Γ be a finite digraph and let G be a subgroup of the automorphism group of Γ. A directed cycle [InlineMediaObject not available: see fulltext.] of Γ is called G-consistent whenever there is an element of G whose restriction to [InlineMediaObject not available: see fulltext.] is the 1-step rotation of [InlineMediaObject not available: see fulltext.]. Consistent cycles in finite arc-transitive graphs were introduced by J. H. Conway in his public lectures at the Second British Combinatorial Conference in 1971. He observed that the number of G-orbits of G-consistent cycles of an arc-transitive group G is precisely one less than the valency of the graph. In this paper, we give a detailed proof of this result in a more general setting of arbitrary groups of automorphisms of graphs and digraphs.

Original language | English (US) |
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Pages (from-to) | 205-216 |

Number of pages | 12 |

Journal | Graphs and Combinatorics |

Volume | 23 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2007 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Graphs and Combinatorics*,

*23*(2), 205-216. https://doi.org/10.1007/s00373-007-0695-2