## Abstract

A minimum feedback arc set of a directed graph G is a smallest set of arcs whose removal makes G acyclic. Its cardinality is denoted by β(G). We show that a simple Eulerian digraph with n vertices and m arcs has β(G) ≥ m ^{2}/2n ^{2}+m/2n, and this bound is optimal for infinitely many m, n. Using this result we prove that a simple Eulerian digraph contains a cycle of length at most 6n ^{2}/m, and has an Eulerian subgraph with minimum degree at least m ^{2}/24n ^{3}. Both estimates are tight up to a constant factor. Finally, motivated by a conjecture of Bollobás and Scott, we also show how to find long cycles in Eulerian digraphs.

Original language | English (US) |
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Pages (from-to) | 859-873 |

Number of pages | 15 |

Journal | Combinatorics Probability and Computing |

Volume | 22 |

Issue number | 6 |

DOIs | |

State | Published - Nov 2013 |

## Keywords

- 2010 Mathematics subject classification: Primary 05C20
- Secondary 05C38