The Minimally Non-Ideal Binary Clutters with a Triangle

Abstract

It is proved that the lines of the Fano plane and the odd circuits of K5 constitute the only minimally non-ideal binary clutters that have a triangle.

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

References

  1. [1]

    A. Abdi and B. Guenin: The two-point Fano and ideal binary clutters, submitted.

  2. [2]

    W. G. Bridges and H. J. Ryser: Combinatorial designs and related systems, J. Algebra 13 (1969), 432–446.

    MathSciNet  Article  Google Scholar 

  3. [3]

    C. Chun and J. Oxley: Internally 4-connected binary matroids with every element in three triangles, preprint, arXiv:1608.06013, (2016)

    Google Scholar 

  4. [4]

    G. Cornuéjols and B. Guenin: Ideal binary clutters, connectivity, and a conjecture of Seymour, SIAM J. Discrete Math. 15 (2002), 329–352.

    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]

    B. Guenin: A characterization of weakly bipartite graphs, J. Combin. Theory Ser. B 83 (2001), 112–168.

    MathSciNet  Article  Google Scholar 

  7. [7]

    B. Guenin: Integral polyhedra related to even-cycle and even-cut matroids, Math. Oper. Res. 27 (2002), 693–710.

    MathSciNet  Article  Google Scholar 

  8. [8]

    B. Guenin, I. Pivotto, P. Wollan: Stabilizer theorems for even cycle matroids, J. Combin. Theory Ser. B 118 (2016), 44–75.

    MathSciNet  Article  Google Scholar 

  9. [9]

    A. Lehman: A solution of the Shannon switching game, Society for Industrial Appl. Math. 12 (1964), 687–725.

    MathSciNet  Article  Google Scholar 

  10. [10]

    A. Lehman: The width-length inequality and degenerate projective planes, DIMACS. 1 (1990), 101–105.

    MathSciNet  MATH  Google Scholar 

  11. [11]

    C. Lütolf and F. Margot: A catalog of minimally nonideal matrices, Math. Methods of Oper. Res. 47 (1998), 221–241.

    MathSciNet  Article  Google Scholar 

  12. [12]

    B. Novick and A. Sebő: On combinatorial properties of binary spaces, IPCO 4 (1995), 212–227.

    MathSciNet  Google Scholar 

  13. [13]

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

    Book  Google Scholar 

  14. [14]

    I. Pivotto: Even cycle and even cut matroids, Ph.D. dissertation, University of Waterloo, 2011.

    Google Scholar 

  15. [15]

    A. Schrijver: A short proof of Guenin's characterization of weakly bipartite graphs, J. Combin. Theory Ser. B 85 (2002), 255–260.

    MathSciNet  Article  Google Scholar 

  16. [16]

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

    MathSciNet  Article  Google Scholar 

  17. [17]

    P. D. Seymour: On Lehman's width-length characterization, DIMACS 1 (1990), 107–117.

    MathSciNet  MATH  Google Scholar 

  18. [18]

    P. D. Seymour: The forbidden minors of binary matrices, J. London Math. Society 2 (1976), 356–360.

    Article  Google Scholar 

  19. [19]

    P. D. Seymour: The matroids with the max-ow min-cut property, J. Combin. Theory Ser. B 23 (1977), 189–222.

    Article  Google Scholar 

  20. [20]

    P. D. Seymour: Triples in matroid circuits. Europ. J. Combinatorics 7 (1986), 177–185.

    MathSciNet  Article  Google Scholar 

  21. [21]

    C. H. Shih: On graphic subspaces of graphic spaces, Ph.D. Dissertation, Ohio State University, 1982.

    Google Scholar 

  22. [22]

    H. Whitney: 2-isomorphic graphs, Amer. J. Math. 55 (1933), 245–254.

    MathSciNet  Article  Google Scholar 

  23. [23]

    T. Zaslavsky: Signed graphs, Discrete Appl. Math. 4 (1982), 47–74.

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ahmad Abdi.

Additional information

This work is supported by NSERC CGS and Discovery grants and by U.S. Office of Naval Research grants under award numbers N00014-15-1-2171 and N00014-18-1-2078.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Abdi, A., Guenin, B. The Minimally Non-Ideal Binary Clutters with a Triangle. Combinatorica 39, 719–752 (2019). https://doi.org/10.1007/s00493-018-3708-2

Download citation

Mathematics Subject Classification (2010)

  • 90C57
  • 05B35