## Abstract

What must one do in order to make acyclic a given oriented graph? Here we look at the structures that must be removed (or reversed) in order to make acyclic a given oriented graph. For a directed acyclic graph H and an oriented graph G, let f_{H}(G) be the maximum number of pairwise disjoint copies of H that can be found in all feedback arc sets of G. In particular, to make G acyclic, one must at least remove (or reverse) f_{H}(G) pairwise disjoint copies of H. Perhaps most intriguing is the case where H is a k-clique, in which case the parameter is denoted by f_{k}(G). Determining f_{k}(G) for arbitrary G seems challenging. Here we essentially answer the problem, precisely, for the family of k-partite tournaments. Let s(G) denote the size of the smallest vertex class of a k-partite tournament G. It is not difficult to show that f_{k}(G) ≤ s(G) - k + 1 (assume that s(G) ≥ k - 1). Our main result is that for all sufficiently large s = s(G), there are k-partite tournaments for which f_{k}(G) = s(G) - k + 1. In fact, much more can be said: A random k-partite tournament G satisfies f_{k}(G) = s(G) - k + 1 almost surely (i.e., with probability tending to 1 as s(G) goes to infinity). In particular, as the title states, f_{k}(G) =[ n/k] - k + 1 almost surely, where G is a random orientation of the Tur an graph T(n, k).

Original language | English |
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Pages (from-to) | 1460-1477 |

Number of pages | 18 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 35 |

Issue number | 2 |

DOIs | |

State | Published - 2021 |

### Bibliographical note

Funding Information:\ast Received by the editors July 30, 2020; accepted for publication (in revised form) March 21, 2021; published electronically June 22, 2021. https://doi.org/10.1137/20M1356506 Funding: This research was supported by the Israel Science Foundation, grant 1082/16. \dagger Department of Mathematics, University of Haifa, Haifa 31905, Israel (nassarsafwat@gmail.com, raphy@math.haifa.ac.il).

Publisher Copyright:

© 2021 Society for Industrial and Applied Mathematics

## Keywords

- Feedback arc set
- K-clique
- Random graph
- Turn tournament

## ASJC Scopus subject areas

- Mathematics (all)