## Abstract

We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph F, denote by mader_{χ→}(F) the smallest integer k such that every k-dichromatic digraph contains a subdivision of F. As our first main result, we prove that if F is an orientation of a cycle then mader_{χ→}(F)=v(F). This settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura and Thomassé. We also extend this result to the more general class of orientations of cactus graphs, and to bioriented forests. Our second main result is that mader_{χ→}(F)=4 for every tournament F of order 4. This is an extension of the classical result by Dirac that 4-chromatic graphs contain a K_{4}-subdivision to directed graphs.

Original language | English |
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Pages (from-to) | 1-30 |

Number of pages | 30 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 153 |

DOIs | |

State | Published - Mar 2022 |

## Keywords

- Dichromatic number
- Digraph
- Subdivision
- Topological minor