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

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

**Consistent cycles in graphs and digraphs.** / Miklavič, Štefko; Potočnik, Primož; Wilson, Stephen E.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 23, no. 2, pp. 205-216. https://doi.org/10.1007/s00373-007-0695-2

}

TY - JOUR

T1 - Consistent cycles in graphs and digraphs

AU - Miklavič, Štefko

AU - Potočnik, Primož

AU - Wilson, Stephen E

PY - 2007/4

Y1 - 2007/4

N2 - 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.

AB - 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=34247253385&partnerID=8YFLogxK

U2 - 10.1007/s00373-007-0695-2

DO - 10.1007/s00373-007-0695-2

M3 - Article

VL - 23

SP - 205

EP - 216

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 2

ER -