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

@article{Huang2013LargeFA,
  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},
  year={2013},
  volume={22},
  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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References

SHOWING 1-10 OF 19 REFERENCES

Directed Graphs Without Short Cycles

TLDR
It is proved that every directed graph whose shortest directed cycle has length at least r ≥ 4 satisfies β(G) ≤ cγ(G)/r2, where c is an absolute constant.

Cycles in dense digraphs

TLDR
It is proved that in general β(G) ≤ γ(G), and that in two special cases: when V (G) is the union of two cliques when the vertices of G can be arranged in a circle such that if distinct u, v, w are in clockwise order and uw is a (directed) edge, then so are both uv, vw.

A Proof of a Conjecture of Bondy Concerning Paths in Weighted Digraphs

Our aim in this note is to prove a conjecture of Bondy, extending a classical theorem of Dirac to edge-weighted digraphs: if every vertex has out-weight at least 1 then the digraph contains a path of

The Caccetta-Haggkvist conjecture and additive number theory

The Caccetta-Haggkvist conjecture states that if G is a finite directed graph with at least n/k edges going out of each vertex, then G contains a directed cycle of length at most k. Hamidoune used

Ranking Tournaments

  • N. Alon
  • Mathematics
    SIAM J. Discret. Math.
  • 2006
TLDR
It is shown that the feedback arc set problem for tournaments is NP-hard under randomized reductions, which settles a conjecture of Bang-Jensen and Thomassen.

The Minimum Feedback Arc Set Problem is NP-Hard for Tournaments

TLDR
It is proved that the minimum feedback arc set problem is NP-hard for tournaments.

Retiming synchronous circuitry

This paper describes a circuit transformation calledretiming in which registers are added at some points in a circuit and removed from others in such a way that the functional behavior of the circuit

A Summary of Results and Problems Related to the Caccetta-Haggkvist Conjecture

This paper is an attempt to survey the current state of our knowledge on the Caccetta-Häggkvist conjecture and related questions. In January 2006 there was a workshop hosted by the American Institute

The logical design of operating systems

TLDR
The logical design of operating systems is discussed in more detail in the second part of this monograph on the design of mobile operating systems.

Modern Graph Theory