Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-23T14:33:10.586Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  12 September 2013

HAO HUANG ,
JIE MA ,
ASAF SHAPIRA ,
BENNY SUDAKOV  and
RAPHAEL YUSTER
HAO HUANG
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095, USA (e-mail: [email protected])
JIE MA
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095, USA (e-mail: [email protected])
ASAF SHAPIRA
Affiliation:
School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected])
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH, 8092 Zurich, Switzerland and Department of Mathematics, UCLA, Los Angeles, CA 90095, USA (e-mail: [email protected])
RAPHAEL YUSTER
Affiliation:
Department of Mathematics, University of Haifa, Haifa 31905, Israel (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

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) ≥ 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.

Keywords

Primary 05C20Secondary 05C38
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 6 , November 2013 , pp. 859 - 873
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alon, N. (2006) Ranking tournaments. SIAM J. Discrete Math. 20 137–142.Google Scholar
[2]Bollobás, B. (1998) Modern Graph Theory, Vol. 184 of Graduate Texts in Mathematics, Springer.Google Scholar
[3]Bollobás, B. and Scott, A. (1996) A proof of a conjecture of Bondy concerning paths in weighted digraphs. J. Combin. Theory Ser. B 66 283–292.Google Scholar
[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 181–187.Google Scholar
[5]Charbit, P., Thomassé, S. and Yeo, A. (2007) The minimum feedback arc set problem is NP-hard for tournaments. Combin. Probab. Comput. 16 1–4.Google Scholar
[6]Chudnovsky, M., Seymour, P. and Sullivan, B. (2008) Cycles in dense digraphs. Combinatorica 28 1–18.CrossRefGoogle Scholar
[7]Fox, J., Keevash, P. and Sudakov, B. (2010) Directed graphs without short cycles. Combin. Probab. Comput. 19 285–301.Google Scholar
[8]Leiserson, C. and Saxe, J. (1991) Retiming synchronous circuitry. Algorithmica 6 5–35.Google Scholar
[9]Nathanson, M. The Caccetta–Häggkvist conjecture and additive number theory. www.aimath.org/preprints.htmlGoogle Scholar
[10]Shaw, A. (1974) The Logical Design of Operating Systems, Prentice Hall.Google Scholar
[11]Sullivan, B. (2008) Extremal problems in digraphs. PhD thesis, Princeton University.Google Scholar
[12]Sullivan, B. A summary of results and problems related to the Caccetta–Häggkvist conjecture. http://arxiv.org/abs/math/0605646v1Google Scholar