### Resumen

A famous conjecture of Gyárfás and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star S and integer k, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size k or it contains S as an induced subgraph. As an evidence, we prove that for any oriented star S, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order 3 or S as an induced subdigraph. We then study for which sets P of orientations of P_{4} (the path on four vertices) similar statements hold. We establish some positive and negative results.

Idioma original | Inglés |
---|---|

Páginas (desde-hasta) | 304-326 |

Número de páginas | 23 |

Publicación | Journal of Graph Theory |

Volumen | 89 |

N.º | 3 |

DOI | |

Estado | En prensa - 1 ene 2018 |

### Áreas temáticas de ASJC Scopus

- Geometría y topología

## Huella Profundice en los temas de investigación de 'χ-bounded families of oriented graphs'. En conjunto forman una huella única.

## Citar esto

*Journal of Graph Theory*,

*89*(3), 304-326. https://doi.org/10.1002/jgt.22252