Large Feedback Arc Sets, High Minimum Degree Subgraphs, and Long Cycles in Eulerian Digraphs

  title={Large Feedback Arc Sets, High Minimum Degree Subgraphs, and Long Cycles in Eulerian Digraphs},
  author={Hao-wei Huang and Jie Ma and Asaf Shapira and Benny Sudakov and Raphael Yuster},
  journal={Combinatorics, Probability and Computing},
  pages={859 - 873}
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) ≥ m2/2n2+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 6n2/m, and has an Eulerian subgraph with minimum degree at least m2/24n3. Both estimates are tight up to a constant factor… 

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