### Abstract

A cycle decomposition of a graph Γ is a set C of cycles of Γ such that every edge of Γ belongs to exactly one cycle in C. Such a decomposition is called arc-transitive if the group of automorphisms of Γ that preserve C setwise acts transitively on the arcs of Γ. In this paper, we study arc-transitive cycle decompositions of tetravalent graphs. In particular, we are interested in determining and enumerating arc-transitive cycle decompositions admitted by a given arc-transitive tetravalent graph. Among other results we show that a connected tetravalent arc-transitive graph is either 2-arc-transitive, or is isomorphic to the medial graph of a reflexible map, or admits exactly one cycle structure.

Original language | English (US) |
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Pages (from-to) | 1181-1192 |

Number of pages | 12 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 98 |

Issue number | 6 |

DOIs | |

State | Published - Nov 1 2008 |

### Keywords

- Automorphism group
- Consistent cycle
- Cycle decomposition
- Graph
- Medial maps

### ASJC Scopus subject areas

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

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## Cite this

*Journal of Combinatorial Theory. Series B*,

*98*(6), 1181-1192. https://doi.org/10.1016/j.jctb.2008.01.005