### 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 | English |
---|---|

Publicación | Journal of Graph Theory |

DOI | |

Estado | Accepted/In press - 1 ene 2018 |

### Huella dactilar

### ASJC Scopus subject areas

- Geometry and Topology

### Citar esto

*Journal of Graph Theory*. https://doi.org/10.1002/jgt.22252

}

*Journal of Graph Theory*. https://doi.org/10.1002/jgt.22252

**χ-bounded families of oriented graphs.** / Aboulker, P.; Bang-Jensen, J.; Bousquet, N.; Charbit, P.; Havet, F.; Maffray, F.; Zamora, J.

Resultado de la investigación: Article

TY - JOUR

T1 - χ-bounded families of oriented graphs

AU - Aboulker, P.

AU - Bang-Jensen, J.

AU - Bousquet, N.

AU - Charbit, P.

AU - Havet, F.

AU - Maffray, F.

AU - Zamora, J.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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 P4 (the path on four vertices) similar statements hold. We establish some positive and negative results.

AB - 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 P4 (the path on four vertices) similar statements hold. We establish some positive and negative results.

KW - Chromatic number

KW - Clique number

KW - χ-bounded

UR - http://www.scopus.com/inward/record.url?scp=85045122375&partnerID=8YFLogxK

U2 - 10.1002/jgt.22252

DO - 10.1002/jgt.22252

M3 - Article

AN - SCOPUS:85045122375

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

ER -