### Abstract

This is the first in the series of articles stemming from a systematical investigation of finite edge-transitive tetravalent graphs, undertaken recently by the authors. In this article, we study a special but important case in which the girth of such graphs is at most 4. In particular, we show that, except for a single arc-transitive graph on 14 vertices, every edge-transitive tetravalent graph of girth at most 4 is the skeleton of an arc-transitive map on the torus or has one of these two properties: (1)there exist two vertices sharing the same neighbourhood,(2)every edge belongs to exactly one girth cycle. Graphs with property (1) or (2) are then studied further. It is shown that they all arise either as subdivided doubles of smaller arc-transitive tetravalent graphs, or as line graphs of triangle-free (G, 1)-regular and (G, 2)-arc-transitive cubic graphs, or as partial line graphs of certain cycle decompositions of smaller tetravalent graphs.

Original language | English (US) |
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Pages (from-to) | 217-236 |

Number of pages | 20 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 97 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2007 |

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### Keywords

- Automorphism group
- Cycle decomposition
- Edge-transitive graphs
- Graph
- Linking ring structure
- Locally arc-transitive graph
- Semisymmetric graph
- Symmetry
- Tetravalent graphs

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory. Series B*,

*97*(2), 217-236. https://doi.org/10.1016/j.jctb.2006.03.007