Internally 4-Connected Binary Matroids with Every Element in Three Triangles


Let M be an internally 4-connected binary matroid with every element in exactly three triangles. Then M has at least four elements e such that si(M/e) is internally 4-connected. This technical result is a crucial ingredient in Abdi and Guenin’s theorem determining the minimally non-ideal binary clutters that have a triangle.

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


  1. [1]

    A. Abdi and B. Guenin: The minimally non-ideal binary clutters with a triangle, Combinatorial, submitted.

  2. [2]

    T. H. Brylawski: A decomposition of combinatorial geometries, Trans. Amer. Math. Soc. 171 (1972), 235–282.

    MathSciNet  Article  Google Scholar 

  3. [3]

    C. Chun, D. Mayhew and J. Oxley: A chain theorem for internally 4-connected binary matroids, J. Combin. Theory Ser. B 101 (2011), 141–189.

    MathSciNet  Article  Google Scholar 

  4. [4]

    G. Cornuéjols and B. Guenin: Ideal binary clutters, connectivity, and a conjecture of Seymour, SIAM J. Discrete Math. 15 (2002), 229–352.

    MathSciNet  Article  Google Scholar 

  5. [5]

    J. Edmonds and D. R. Fulkerson: Bottleneck extrema, J. Combin. Theory Ser. B 8 (1970), 299–306.

    MathSciNet  Article  Google Scholar 

  6. [6]

    A. P. Heron: Some topics in matroid theory, D. Phil. thesis, University of Oxford, 1972.

    Google Scholar 

  7. [7]

    J. Oxley: Matroid theory, Second edition, Oxford University Press, New York, 2011.

    Google Scholar 

  8. [8]

    J. Oxley, C. Semple and G. Whittle: The structure of the 3-separations of 3-connected matroids, J. Combin. Theory Ser. B 92 (2004), 257–293.

    MathSciNet  Article  Google Scholar 

  9. [9]

    H. Qin and X. Zhou: The class of binary matroids with no M(K 3,3)-, M*(K 3,3)-, M(K 5)-, or M*(K 5)-minor, J. Combin. Theory Ser. B 90 (2004), 173–184.

    MathSciNet  Article  Google Scholar 

  10. [10]

    P.D. Seymour: The forbidden minors of binary clutters, J. London Math. Soc (2) 12 (1975/76), 356–360.

    MathSciNet  MATH  Google Scholar 

  11. [11]

    P.D. Seymour: The matroids with the max-flow min-cut property, J. Combin. Theory Ser. B 23 (1977), 199–222.

    MathSciNet  Article  Google Scholar 

  12. [12]

    P.D. Seymour: Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305–359.

    MathSciNet  Article  Google Scholar 

  13. [13]

    P. D. Seymour: Matroids and multicommodity flows, Europ. J. Combin. 2 (1981), 257–290.

    MathSciNet  Article  Google Scholar 

  14. [14]

    W. T. Tutte: Connectivity in matroids, Canad. J. Math. 18 (1966), 1301–1324.

    MathSciNet  Article  Google Scholar 

Download references


The authors thank the referees for numerous suggestions that both corrected errors and improved the exposition.

Author information



Corresponding authors

Correspondence to Carolyn Chun or James Oxley.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chun, C., Oxley, J. Internally 4-Connected Binary Matroids with Every Element in Three Triangles. Combinatorica 39, 825–845 (2019).

Download citation

Mathematics Subject Classification (2010)

  • 05B35
  • 52B40