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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.

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[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.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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