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

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 |

### Fingerprint

### 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

**Tetravalent edge-transitive graphs of girth at most 4.** / Potočnik, Primož; Wilson, Stephen E.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 97, no. 2, pp. 217-236. https://doi.org/10.1016/j.jctb.2006.03.007

}

TY - JOUR

T1 - Tetravalent edge-transitive graphs of girth at most 4

AU - Potočnik, Primož

AU - Wilson, Stephen E

PY - 2007/3

Y1 - 2007/3

N2 - 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.

AB - 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.

KW - Automorphism group

KW - Cycle decomposition

KW - Edge-transitive graphs

KW - Graph

KW - Linking ring structure

KW - Locally arc-transitive graph

KW - Semisymmetric graph

KW - Symmetry

KW - Tetravalent graphs

UR - http://www.scopus.com/inward/record.url?scp=33846160597&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846160597&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2006.03.007

DO - 10.1016/j.jctb.2006.03.007

M3 - Article

AN - SCOPUS:33846160597

VL - 97

SP - 217

EP - 236

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 2

ER -