Advertisement

Recent advances

  • P. D. Seymour
  • J. Kahn
  • J. P. S. Kung

Abstract

In the final chapter of this anthology, we reprint two recent papers which show that the possibilities in matroid theory may not be as wide as has been thought.

Keywords

Projective Geometry Universal Model Exceptionable Point Combinatorial Geometry Series Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bixby, R.E.: Kuratowski’s and Wagner’s theorems for matroids, J. Combin. Theory Ser. B 22(1977), 31–53.MathSciNetCrossRefMATHGoogle Scholar
  2. Bixby, R.E.: A simple theorem on 3-connectivity, Linear Algebra Appl. 45(1982), 123–126.MathSciNetCrossRefMATHGoogle Scholar
  3. Duffin, R.J.: Topology of series-parallel networks, J. Math. Anal. Appl. 10 (1965), 303–318.MathSciNetCrossRefMATHGoogle Scholar
  4. Ore, O.: The four-color problem, Academic Press, New York and London, 1967.MATHGoogle Scholar
  5. Seymour, P.D.: The matroids with the max-flow min-cut property, J. Combin. Theory Ser. B 23(1977), 189–222.MathSciNetCrossRefMATHGoogle Scholar
  6. Seymour, P.D.: On Tutte’s characterization of graphic matroids, Combinatorics 79, Part I, Ann. Discrete Math. 8(1980), 83–90.MathSciNetCrossRefMATHGoogle Scholar
  7. Seymour, P.D.: Some applications of matroid decomposition, Algebraic Methods in Graph Theory, Vol. I, II (Proc. Conf., Szeged, 1978), pp. 713–726, Colloq. Math. Soc. János Bolyai 25, North-Holland, Amsterdam, 1981.Google Scholar
  8. Seymour, P.D.: On Tutte’s extension of the four-colour problem, J. Combin. Theory Ser. B 31(1981), 82–94.MathSciNetCrossRefMATHGoogle Scholar
  9. Seymour, P.D.: Matroids and multicommodity flows, Europ. J. Combin. 2 (1981), 257–290.MathSciNetMATHGoogle Scholar
  10. Truemper, K.: A decomposition theory for matroids. I: General results, J. Combin. Theory Ser. B 39(1985), 43–76.MathSciNetCrossRefMATHGoogle Scholar
  11. Truemper, K.: A decomposition theory for matroids. II: Minimal violation matroids, J. Combin. Theory Ser. B 39(1985), 282–297.MathSciNetCrossRefMATHGoogle Scholar
  12. Truemper, K.: A decomposition theory for matroids. III: Decomposition conditions, preprint.Google Scholar
  13. Wagner, K.: Bemerkungen zu Hadwigers Vermutung, Math. Ann. 141 (1960), 433–451.MathSciNetCrossRefMATHGoogle Scholar
  14. Walton, P.N. and Welsh, D.J.A.: On the chromatic number of binary matroids, Mathematika 27(1980), 1–9.MathSciNetCrossRefMATHGoogle Scholar
  15. Birkhoff, G.: On the structure of abstract algebras, Proc. Cambridge Philos. Soc. 31(1935), 433–454.CrossRefGoogle Scholar
  16. Ingleton, A.W.: Transversal matroids and related structures, Higher Combinatorics (M. Aigner, ed.), pp. 117–131, Reidel, Dordrecht, 1977.CrossRefGoogle Scholar
  17. Kahn, J. and Kung, J.P.S.: A classification of modularly complemented lattices, Europ. J. Combin., to appear.Google Scholar
  18. Kung, J.P.S.: Numerically regular hereditary classes of combinatorial geometries, Geometriae Dedicata, to appear.Google Scholar
  19. Sims, J.A.: A complete class of matroids, Quart. J. Math. Oxford Ser. (2) 28 (1977), 449–451.MathSciNetCrossRefMATHGoogle Scholar
  20. 1.
    R. E. Bixby, Kuratowski’s and Wagner’s theorems for matroids, J. Combinatorial Theory Ser. B 22 (1977), 31–53.MathSciNetCrossRefMATHGoogle Scholar
  21. 2.
    R. E. Bixby and W. H. Cunningham, Matroids, graphs, and 3-connectivity, in “Graph Theory and Related Topics” (J. A. Bondy and U. S. R. Murty, Eds.), pp. 91–103, Academic Press, New York, 1979.Google Scholar
  22. 3.
    T. Brylawski, A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971), 1–22.MathSciNetCrossRefMATHGoogle Scholar
  23. 4.
    T. Brylawski, Modular constructions for combinatorial geometries, Trans. Amer. Math. Soc. 203 (1975), 1–44.MathSciNetCrossRefMATHGoogle Scholar
  24. 5.
    W. H. Cunningham, “A Combinatorial Decomposition Theory,” Thesis (Research Report CORR 74-13), University of Waterloo, 1973.Google Scholar
  25. 6.
    W. H. Cunningham, On matroid connectivity, to appear.Google Scholar
  26. 7.
    A. K. Kelmans, A theory of 3-connected graphs, in “Proceedings, Trudi Colloquium on Algebraic Methods in Graph Theory,” Szeged, Hungary, in press.Google Scholar
  27. 8.
    P. D. Seymour, Matroids and multicommodity flows, European J. Combinatorics, in press.Google Scholar
  28. 9.
    P. D. Seymour, On Tutte’s characterization of graphic matroids, in “Combinatorics 1979,” Annals of Discrete Mathematics, North-Holland, Amsterdam, in press.Google Scholar
  29. 10.
    P. D. Seymour, On Tutte’s extension of the four-colour problem, J. Combinatorial Theory Ser. B, in press.Google Scholar
  30. 11.
    W. T. Tutte, A homotopy theorem for matroids, I, II, Trans. Amer. Math. Soc. 88 (1958), 144–174.MathSciNetMATHGoogle Scholar
  31. 12.
    W. T. Tutte, “Connectivity in Graphs,” Univ. of Toronto Press, Toronto, 1966.Google Scholar
  32. 13.
    W. T. Tutte, Connectivity in matroids, Canad. J. Math. 18 (1966), 1301–1324.MathSciNetCrossRefMATHGoogle Scholar
  33. 14.
    W. T. Tutte, Matroids and graphs, Trans. Amer. Math. Soc. 90 (1959), 527–552.MathSciNetCrossRefMATHGoogle Scholar
  34. 15.
    K. Wagner, “Beweis einer Abschwächung der Hadwiger-Vermutung,” Math. Ann. 153 (1964), 139–141.MathSciNetCrossRefMATHGoogle Scholar
  35. 16.
    D. J. A. Welsh, “Matroid Theory,” Academic Press, New York, 1976.Google Scholar
  36. 1.
    G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc. 31 (1935), 433–454.CrossRefGoogle Scholar
  37. 2.
    G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R.I., 1967.Google Scholar
  38. 3.
    P. M. Cohn, Universal algebra, Harper and Row, New York, 1965.MATHGoogle Scholar
  39. 4.
    H. H. Crapo and G.-C. Rota, On the foundations of combinatorial theory: Combinatorial geometries (preliminary edition), M.I.T. Press, Cambridge, 1970.MATHGoogle Scholar
  40. 5.
    P. Doubilet, G.-C. Rota and R. Stanley, On the foundations of combinatorial theory. VI: The idea of generating function, Sixth Berkeley Sympos. on Math. Statist, and Prob., Vol. II: Probability Theory, Univ. California, Berkeley, Calif., 1972, pp. 267–318.Google Scholar
  41. 6.
    T. A. Dowling, A q-analog of the partition lattice, A Survey of Combinatorial Theory (J. N. Srivastava, editor), North-Holland, Amsterdam, 1973, pp. 101–115.Google Scholar
  42. 7.
    T. A. Dowling, A class of geometric lattices based on finite groups, J. Combin. Theory Ser. B 14 (1973), 61–86.MathSciNetCrossRefMATHGoogle Scholar
  43. 8.
    C. Greene, Lectures in combinatorial geometries, Notes from the NSF Seminar in Combinatorial Theory, Bowdoin College, 1971, unpublished.Google Scholar
  44. 9.
    S. Mac Lane, A lattice formulation for transcendence degrees and p-bases, Duke Math. J. 4 (1938), 455–468.MathSciNetCrossRefMATHGoogle Scholar
  45. 10.
    D. Kelly and G.-C. Rota, Some problems in combinatorial geometry, A Survey of Combinatorial Theory (J. N. Srivastava, editor), North-Holland, Amsterdam, 1973, pp. 309–312.Google Scholar
  46. 11.
    R. P. Stanley, Modular elements of geometric lattices, Algebra Universalis 1 (1971), 214–217.MathSciNetCrossRefMATHGoogle Scholar
  47. 12.
    R. P. Stanley, Supersolvable lattices, Algebra Universalis 2 (1972), 197–217.MathSciNetCrossRefMATHGoogle Scholar
  48. 13.
    D. J. A. Welsh, Matroid theory, Academic Press, London, 1976.MATHGoogle Scholar
  49. 14.
    T. Zaslavsky, Biased graphs, preprint, 1977.Google Scholar
  50. 15.
    T. Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982), 47–74.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • P. D. Seymour
    • 1
    • 2
  • J. Kahn
    • 3
    • 4
  • J. P. S. Kung
    • 3
    • 4
  1. 1.Merton CollegeOxfordEngland
  2. 2.University of WaterlooWaterlooCanada
  3. 3.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  4. 4.Department of MathematicsNorth Texas State UniversityDentonUSA

Personalised recommendations