### Abstract

It is shown that the automorphism group of an infinite, locally finite, planar graph acts primitively on its vertex set if and only if the graph has connectivity 1 and, for some integer m≥2, every vertex is incident with exactly m lobes, all of which are finite. Specifically, either all of the lobes are isomorphic to K_{4} or all are circuits of length p for some odd prime p.

Original language | English (US) |
---|---|

Pages (from-to) | 87-104 |

Number of pages | 18 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 91 |

Issue number | 1 |

DOIs | |

State | Published - May 2004 |

### Fingerprint

### Keywords

- Edge-transitive
- Infinite planar map
- Primitive permutation group

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory, Series B*,

*91*(1), 87-104. https://doi.org/10.1016/j.jctb.2003.10.005

**A characterization of infinite planar primitive graphs.** / Watkins, Mark E.; Graver, Jack E.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series B*, vol. 91, no. 1, pp. 87-104. https://doi.org/10.1016/j.jctb.2003.10.005

}

TY - JOUR

T1 - A characterization of infinite planar primitive graphs

AU - Watkins, Mark E.

AU - Graver, Jack E

PY - 2004/5

Y1 - 2004/5

N2 - It is shown that the automorphism group of an infinite, locally finite, planar graph acts primitively on its vertex set if and only if the graph has connectivity 1 and, for some integer m≥2, every vertex is incident with exactly m lobes, all of which are finite. Specifically, either all of the lobes are isomorphic to K4 or all are circuits of length p for some odd prime p.

AB - It is shown that the automorphism group of an infinite, locally finite, planar graph acts primitively on its vertex set if and only if the graph has connectivity 1 and, for some integer m≥2, every vertex is incident with exactly m lobes, all of which are finite. Specifically, either all of the lobes are isomorphic to K4 or all are circuits of length p for some odd prime p.

KW - Edge-transitive

KW - Infinite planar map

KW - Primitive permutation group

UR - http://www.scopus.com/inward/record.url?scp=2342511080&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2342511080&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2003.10.005

DO - 10.1016/j.jctb.2003.10.005

M3 - Article

AN - SCOPUS:2342511080

VL - 91

SP - 87

EP - 104

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -