## Abstract

For an oriented graph G, let f(G) denote the maximum chromatic number of an acyclic subgraph of G. Let f(n) be the smallest integer such that every oriented graph G with chromatic number larger than f(n) has f(G)>n. Let g(n) be the smallest integer such that every tournament G with more than g(n) vertices has f(G)>n. It is straightforward that Ω(n)≤g(n)≤f(n)≤n^{2}. This paper provides the first nontrivial lower and upper bounds for g(n). In particular, it is proved that [Formula presented]n^{8∕7}≤g(n)≤n^{2}−(2−[Formula presented])n+2. It is also shown that f(2)=3, i.e. every orientation of a 4-chromatic graph has a 3-chromatic acyclic subgraph. Finally, it is shown that a random tournament G with n vertices has f(G)=Θ([Formula presented]) whp.

Original language | English |
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Pages (from-to) | 11-18 |

Number of pages | 8 |

Journal | European Journal of Combinatorics |

Volume | 75 |

DOIs | |

State | Published - Jan 2019 |

### Bibliographical note

Funding Information:This research was supported by the Israel Science Foundation, Israel (Grant No. 1082/16).

Publisher Copyright:

© 2018 Elsevier Ltd

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics