# Cyclically five-connected cubic graphs

## Author(s): Robertson, Neil; Seymour, Paul D.; Thomas, Robin

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Abstract: | A cubic graph G is cyclically 5-connected if G is simple, 3-connected, has at least 10 vertices and for every set F of edges of size at most four, at most one component of G\textbackslashF contains circuits. We prove that if G and H are cyclically 5-connected cubic graphs and H topologically contains G, then either G and H are isomorphic, or (modulo well described exceptions) there exists a cyclically 5-connected cubic graph G’ such that H topologically contains G’ and G’ is obtained from G in one of the following two ways. Either G’ is obtained from G by subdividing two distinct edges of G and joining the two new vertices by an edge, or G’ is obtained from G by subdividing each edge of a circuit of length five and joining the new vertices by a matching to a new circuit of length five disjoint from G in such a way that the cyclic orders of the two circuits agree. We prove a companion result, where by slightly increasing the connectivity of H we are able to eliminate the second construction. We also prove versions of both of these results when G is almost cyclically 5-connected in the sense that it satisfies the definition except for 4-edge cuts such that one side is a circuit of length four. In this case G’ is required to be almost cyclically 5-connected and to have fewer circuits of length four than G. In particular, if G has at most one circuit of length four, then G’ is required to be cyclically 5-connected. However, in this more general setting the operations describing the possible graphs G’ are more complicated. (C) 2017 Published by Elsevier Inc. |

Publication Date: | Jul-2017 |

Electronic Publication Date: | 31-Mar-2017 |

Citation: | Robertson, Neil, Seymour, PD, Thomas, Robin. (2017). Cyclically five-connected cubic graphs. JOURNAL OF COMBINATORIAL THEORY SERIES B, 125 (132 - 167. doi:10.1016/j.jctb.2017.03.003 |

DOI: | doi:10.1016/j.jctb.2017.03.003 |

ISSN: | 0095-8956 |

EISSN: | 1096-0902 |

Pages: | 132 - 167 |

Type of Material: | Journal Article |

Journal/Proceeding Title: | JOURNAL OF COMBINATORIAL THEORY SERIES B |

Version: | Author's manuscript |

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