### Abstract

In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t+1 vertices is t-colourable. When t≤3 this is easy, and when t=4, Wagner's theorem of 1937 shows the conjecture to be equivalent to the four-colour conjecture (the 4CC). However, when t≥5 it has remained open. Here we show that when t=5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwiger's conjecture when t=5 is "apex", that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwiger's conjecture when t=5, because it implies that apex graphs are 5-colourable.

Original language | English (US) |
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Pages (from-to) | 279-361 |

Number of pages | 83 |

Journal | Combinatorica |

Volume | 13 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 1993 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

_{6}-free graphs.

*Combinatorica*,

*13*(3), 279-361. https://doi.org/10.1007/BF01202354