The gallai-younger conjecture for planar graphs

Abstract

Younger conjectured that for everyk there is ag(k) such that any digraphG withoutk vertex disjoint cycles contains a setX of at mostg(k) vertices such thatG−X has no directed cycles. Gallai had previously conjectured this result fork=1. We prove this conjecture for planar digraphs. Specifically, we show that ifG is a planar digraph withoutk vertex disjoint directed cycles, thenG contains a set of at mostO(klog(k)log(log(k))) vertices whose removal leaves an acyclic digraph. The work also suggests a conjecture concerning an extension of Vizing's Theorem for planar graphs.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    T. Gallai: Problem 6,Theory of Graphs, Proc. Col. Tihany, 1968, 362.

  2. [2]

    B.Bollobás: Graphs without two independent circuits (Hungarian),K. Mat. Lapok,15 (1964), 311–321.

    Google Scholar 

  3. [3]

    C. Hoang andB. Reed: A Note on Short Cycles in Digraphs,Journal of Combinatorial Theory (B),66 (1987), 103–107.

    Google Scholar 

  4. [4]

    W. McCuaig: Intercyclic Digraphs,Graph Structure Theory, Contemporary Mathematics147 (1993), 203–246.

    Google Scholar 

  5. [5]

    Alice Metzlar andU. S. R. Murty: Disjoint Circuits in Planar Digraphs,Manuscript, 1991.

  6. [6]

    P. Seymour: Fractionally Packing Disjoint Circuits,Combinatorica, to appear.

  7. [7]

    C. Thomassen: Disjoint Cycles in Digraphs,Combinatorica,14 (1983), 393–396.

    Google Scholar 

  8. [8]

    Thurston: The geometry and topology of three-manifoldsunpublished.

  9. [9]

    D. Younger: Graphs with interlinked directed circuits,Proceedings of the Midwest Symposium on Circuit Theory 2, (1973), XVI 2.1-XVI 2.7.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Reed, B.A., Shepherd, F.B. The gallai-younger conjecture for planar graphs. Combinatorica 16, 555–566 (1996). https://doi.org/10.1007/BF01271273

Download citation

Mathematics Subject Classification (1991)

  • 05 C 75
  • 05 C 70
  • 90 C 10
  • 90 C 27