### Abstract

A graph is apex if there is a vertex whose deletion makes the graph planar, and doublecross if it can be drawn in the plane with only two crossings, both incident with the infinite region in the natural sense. In 1966, Tutte [9] conjectured that every two-edge-connected cubic graph with no Petersen graph minor is three-edge-colourable. With Neil Robertson, two of us showed that this is true in general if it is true for apex graphs and doublecross graphs [6,7]. In another paper [8], two of us solved the apex case, but the doublecross case remained open. Here we solve the doublecross case; that is, we prove that every two-edge-connected doublecross cubic graph is three-edge-colourable. The proof method is a variant on the proof of the four-colour theorem given in [5].

Original language | English (US) |
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Pages (from-to) | 66-95 |

Number of pages | 30 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 119 |

DOIs | |

State | Published - Jul 1 2016 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Keywords

- Edge-colouring
- Four-colour theorem
- Petersen minor

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## Cite this

*Journal of Combinatorial Theory. Series B*,

*119*, 66-95. https://doi.org/10.1016/j.jctb.2015.12.006