## Bounded Degree Conjecture Holds Precisely for *c*-Crossing-Critical Graphs with *c* ≤ 12

Autor(en): | Bokal, Drago Dvorak, Zdenek Hlineny, Petr Leanos, Jesus Mohar, Bojan Wiedera, Tilo |

Stichwörter: | 05C10; ADDITIVITY; INFINITE FAMILIES; Mathematics; NUMBER |

Erscheinungsdatum: | 2022 |

Herausgeber: | SPRINGER HEIDELBERG |

Enthalten in: | COMBINATORICA |

Band: | 42 |

Ausgabe: | 5 |

Startseite: | 701 |

Seitenende: | 728 |

Zusammenfassung: | We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For every fixed pair of integers with c >= 13 and d >= 1, we give first explicit constructions of c-crossing-critical graphs containing arbitrarily many vertices of degree greater than d. We also show that such unbounded degree constructions do not exist for c <= 12, precisely, that there exists a constant D such that every c-crossing-critical graph with c <= 12 has maximum degree at most D. Hence, the bounded maximum degree conjecture of c-crossing-critical graphs, which was generally disproved in 2010 by Dvorak and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values c <= 12.(1) |

ISSN: | 0209-9683 |

DOI: | 10.1007/s00493-021-4285-3 |

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