Large feedback arc sets, high minimum degree subgraphs, and long cycles in eulerian digraphs

Hao Huang, Jie Ma, Asaf Shapira, Benny Sudakov, Raphael Yuster

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

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 languageEnglish (US)
Pages (from-to)859-873
Number of pages15
JournalCombinatorics Probability and Computing
Volume22
Issue number6
DOIs
StatePublished - Nov 2013

Keywords

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

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