Skip to main content

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


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. Finally, motivated by a conjecture of Bollobás and Scott, we also show how to find long cycles in Eulerian digraphs.

Hide All
[1]Alon, N. (2006) Ranking tournaments. SIAM J. Discrete Math. 20 137142.
[2]Bollobás, B. (1998) Modern Graph Theory, Vol. 184 of Graduate Texts in Mathematics, Springer.
[3]Bollobás, B. and Scott, A. (1996) A proof of a conjecture of Bondy concerning paths in weighted digraphs. J. Combin. Theory Ser. B 66 283292.
[4]Caccetta, L. and Häggkvist, R. (1978) On minimal digraphs with given girth. In Proc. 9th Southeastern Conference on Combinatorics, Graph Theory, and Computing: Boca Raton 1978. Congress. Numer. XXI 181187.
[5]Charbit, P., Thomassé, S. and Yeo, A. (2007) The minimum feedback arc set problem is NP-hard for tournaments. Combin. Probab. Comput. 16 14.
[6]Chudnovsky, M., Seymour, P. and Sullivan, B. (2008) Cycles in dense digraphs. Combinatorica 28 118.
[7]Fox, J., Keevash, P. and Sudakov, B. (2010) Directed graphs without short cycles. Combin. Probab. Comput. 19 285301.
[8]Leiserson, C. and Saxe, J. (1991) Retiming synchronous circuitry. Algorithmica 6 535.
[9]Nathanson, M. The Caccetta–Häggkvist conjecture and additive number theory.
[10]Shaw, A. (1974) The Logical Design of Operating Systems, Prentice Hall.
[11]Sullivan, B. (2008) Extremal problems in digraphs. PhD thesis, Princeton University.
[12]Sullivan, B. A summary of results and problems related to the Caccetta–Häggkvist conjecture.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 21 *
Loading metrics...

Abstract views

Total abstract views: 146 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd July 2018. This data will be updated every 24 hours.