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 |
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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
ASJC Scopus subject areas
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics