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Combinatorica

, 31:55 | Cite as

Destroying cycles in digraphs

  • Molly Dunkum
  • Peter Hamburger
  • Attila Pór
Article

Abstract

For a simple directed graph G with no directed triangles, let β(G) be the size of the smallest subset XE(G) such that G\X has no directed cycles, and let γ(G) denote the number of unordered pairs of nonadjacent vertices in G. Chudnovsky, Seymour, and Sullivan showed that β(G) ≤ γ(G), and conjectured that β(G) ≤ \(\tfrac{{\gamma (G)}} {2}\) . In this paper we prove that β(G)<0.88γ(G).

Mathematics Subject Classification (2000)

05C20 05C38 05C85 

References

  1. [1]
    L. Caccetta and R. Häggkvist: On minimal digraphs with given girth, Congressus Numerantium XXI (1978), 181–187.Google Scholar
  2. [2]
    M. Chudnovsky, P. Seymour and B. Sullivan: Cycles in dense digraphs, Combinatorica 28(1) (2008), 1–18.MathSciNetCrossRefGoogle Scholar
  3. [3]
    J. Fox, P. Keevash and B. Sudakov: Directed graphs without short cycles, Combinatorics, Probability, and Computation 19(2) (2009), 285–301.MathSciNetCrossRefGoogle Scholar
  4. [4]
    P. Hamburger, P. Haxell and A. Kostochka: On directed triangles in digraphs, Electronic Journal of Combinatorics 14 (2007), #N19.MathSciNetGoogle Scholar
  5. [5]
    B. Sullivan: A summary of results and problems related to the Caccetta-Häggkvist conjecture, manuscript, AIM Preprint 2006-13, www.aimath.org/preprints.html.

Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceWestern Kentucky UniversityBowling GreenUSA

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