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

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 2008 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Journal of Combinatorial Theory. Series B*,

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

**Arc-transitive cycle decompositions of tetravalent graphs.** / Miklavič, Štefko; Potočnik, Primož; Wilson, Stephen E.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 98, no. 6, pp. 1181-1192. https://doi.org/10.1016/j.jctb.2008.01.005

}

TY - JOUR

T1 - Arc-transitive cycle decompositions of tetravalent graphs

AU - Miklavič, Štefko

AU - Potočnik, Primož

AU - Wilson, Stephen E

PY - 2008/11

Y1 - 2008/11

N2 - 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.

AB - 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.

KW - Automorphism group

KW - Consistent cycle

KW - Cycle decomposition

KW - Graph

KW - Medial maps

UR - http://www.scopus.com/inward/record.url?scp=52149115363&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=52149115363&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2008.01.005

DO - 10.1016/j.jctb.2008.01.005

M3 - Article

VL - 98

SP - 1181

EP - 1192

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 6

ER -