Advertisement

Linear representation of matroids

  • Saunders Maclane
  • W. T. Tutte
  • P. D. Seymour

Abstract

The axioms of matroid theory are abstracted from the properties of linear dependence. Thus, a natural question is whether every matroid can be represented by a set of vectors with coordinates over some field k so that abstract dependence in the matroid coincides with linear dependence of the vectors. If such a set of vectors exists, the matroid is said to be representable over the field k. This question was first considered by Whitney in [I.1], §16. There he gave the Fano plane as an example of a matroid representable only over a field of characteristic 2, and hence, not representable over the real or complex numbers. The question whether there exists a matroid not representable over any field is answered in Mac Lane’s paper, “Some interpretation of abstract linear dependence in terms of projective geometry,” the first paper reprinted in this chapter.

Keywords

Elementary Chain Connected Line Connected Plane MATROID Representation Fano Plane 
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. Brylawski, T.: Finite prime-field characteristic sets for planar configurations, Linear Algebra Appl. 46(1982), 155–176.MathSciNetMATHCrossRefGoogle Scholar
  2. Dowling, T.A.: A class of geometric lattices based on finite groups, J. Combin. Theory Ser. B 14(1973), 61–86.MathSciNetMATHCrossRefGoogle Scholar
  3. Fournier, J.-C.: Représentation sur un corps des matroïds d’ordre ⩽ 8, Théorie des Matroïdes (Rencontre Franco-Britannique, Brest, 1970), pp. 50–61, Lecture Notes in Math., Vol. 211, Springer-Verlag, Berlin, 1971.CrossRefGoogle Scholar
  4. Fenton, N.E.: Matroid representations — an algebraic treatment, Quart. J. Math. Oxford Ser. (2) 35(1984), 263–280.MathSciNetMATHCrossRefGoogle Scholar
  5. Gordon, G.: Matroids over F p which are rational excluded minors, Discrete Math. 52(1984), 51–65.MathSciNetMATHCrossRefGoogle Scholar
  6. Gordon, G.: Constructing prime-field planar configurations, Proc. Amer. Math. Soc. 91(1984), 492–502.MathSciNetMATHCrossRefGoogle Scholar
  7. Higgs, D.A.: Geometry, Lecture notes, Univ. Waterloo, Waterloo, Ont., 1967.Google Scholar
  8. Ingleton, A.W.: Note on independence functions and rank, J. London Math. Soc. 34(1959), 49–56.MathSciNetMATHCrossRefGoogle Scholar
  9. Ingleton, A.W.: Representations of matroids, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), pp. 149–167, Academic Press, London, 1971.Google Scholar
  10. Ingleton, A.W.: Conditions for representability and transversality of matroids, Théorie des matroïdes (Rencontre Franco-Britannique, Brest, 1970), pp. 62–66, Lecture Notes in Math., Vol. 211, Springer-Verlag, Berlin, 1971.CrossRefGoogle Scholar
  11. Jensen, P.M. and Korte, B.: Complexity of matroid property algorithms, SIAM J. Comput. 11(1982), 184–190.MathSciNetMATHCrossRefGoogle Scholar
  12. Kahn, J.: Characteristic sets of matroids, J. London Math. Soc. (2) 26(1982), 207–217.MathSciNetMATHCrossRefGoogle Scholar
  13. Lazarson, T.: The representation problem for independence functions, J. London Math. Soc. 33(1958), 21–25.MathSciNetMATHCrossRefGoogle Scholar
  14. Mason, J.H.: Geometric realization of combinatorial geometries, Proc. Amer. Math. Soc. 30(1971), 15–21.MathSciNetMATHCrossRefGoogle Scholar
  15. Mason, J.H.: Matroids as the study of geometric configurations, Higher Combinatorics (M. Aigner, ed.), Reidel, Dordrecht, 1977.Google Scholar
  16. Rado, R.: Note on independence functions, Proc. London Math. Soc. (3) 7(1957), 300–320.MathSciNetMATHCrossRefGoogle Scholar
  17. Reid, R.: Obstructions to representations of combinatorial geometries, unpublished; appeared as an appendix to T. Brylawski and D.G. Kelly, Matroids and Combinatorial Geometries, Univ. North Carolina Press, Chapel Hill, N.C., 1980.Google Scholar
  18. Seymour, P.D. and Walton, P.N.: Detecting matroid minors, J. London Math. Soc. (2) 23(1981), 193–203.MathSciNetMATHCrossRefGoogle Scholar
  19. Truemper, K.: On the efficiency of representability tests for matroids, Europ. J. Combin. 3(1982), 275–291.MathSciNetMATHGoogle Scholar
  20. Vámos, P.: Linearity of matroids over division rings (Notes by G. Roulet), Möbius Algebras (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1971), pp. 170–174, Univ. Waterloo, Waterloo, Ont., 1971.Google Scholar
  21. Vámos, P.: A necessary and sufficient condition for a matroid to be linear, Möbius Algebras (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1971), pp. 162–169, Univ. Waterloo, Waterloo, Ont., 1971.Google Scholar
  22. Vámos, P.: The missing axiom of matroid theory is lost forever, J. London Math. Soc. (2) 18(1978), 403–408.MathSciNetMATHCrossRefGoogle Scholar
  23. White, N.L.: The bracket ring of a combinatorial geometry. I and II. Trans. Amer. Math. Soc. 202(1975), 79–95 and 214(1915), 233–248.MathSciNetMATHCrossRefGoogle Scholar
  24. White, N.L.: The transcendence degree of a coordinatization of a combinatorial geometry, J. Combin. Theory Ser. B 29(1980), 168–175.MathSciNetMATHCrossRefGoogle Scholar
  25. Crapo, H.H.: Single-element extensions of matroids, J. Res. Nat. Bur. Standards Sect. B 69B(1965), 55–65.MathSciNetCrossRefGoogle Scholar
  26. Kuratowski, K.: Sur les problèmes des courbes gauches en Topologie, Fundamenta Math. 15(1930), 271–283.MATHGoogle Scholar
  27. Lovász, L.: A homology theory for spanning trees of a graph, Acta Math. Acad. Sci. Hungar. 30(1977), 241–251.MathSciNetCrossRefGoogle Scholar
  28. Maurer, S.B.: Matroid basis graphs. I, J. Combin. Theory Ser. B 14(1973), 216–240.MATHCrossRefGoogle Scholar
  29. Tutte, W.T.: Matroids and graphs, Trans. Amer. Math. Soc. 90(1959), 527–552.MathSciNetMATHCrossRefGoogle Scholar
  30. Bixby, R.E.: ℓ-matrices and a characterization of binary matroids, Discrete Math. 8(1974), 139–145.MathSciNetMATHCrossRefGoogle Scholar
  31. Bixby, R.E.: A strengthened form of Tutte’s characterization of regular matroids, J. Combin. Theory Ser. B 20(1976), 216–221.MathSciNetMATHCrossRefGoogle Scholar
  32. Brylawski, T.: A note on Tutte’s unimodular representation theorem, Proc. Amer. Math. Soc. 52(1975), 499–502.MathSciNetMATHGoogle Scholar
  33. Seymour, P.D.: A note on the production of matroid minors, J. Combin. Theory Ser. B 22(1977), 289–295.MathSciNetMATHCrossRefGoogle Scholar
  34. Seymour, P.D.: On minors of non-binary matroids, Combinatorica 1(1981), 387–394.MathSciNetMATHCrossRefGoogle Scholar
  35. Tutte, W.T.: Lectures on matroids, J. Res. Nat. Bur. Standards Sect. B 69B(1965), 1–47.MathSciNetCrossRefGoogle Scholar
  36. Tutte, W.T.: Introduction to the theory of matroids, American Elsevier, New York, 1971.MATHGoogle Scholar
  37. White, N.L.: Coordinatization of combinatorial geometries, Combinatorial Mathematics and its Applications (Proc. Conf., Univ. North Carolina, Chapel Hill, N.C., 1970), pp. 484–486, Univ. North Carolina, Chapel Hill, N.C., 1970.Google Scholar
  38. Bixby, R.E.: On Reid’s characterization of the ternary matroids, J. Combin. Theory Ser. B 26(1979), 174–204.MathSciNetMATHCrossRefGoogle Scholar
  39. Kahn, J.: A geometric approach to forbidden minors for GF(3), J. Combin. Theory Ser. A 37(1984), 1–12.MathSciNetMATHCrossRefGoogle Scholar
  40. Truemper, K.: Alpha-balanced graphs and matrices and GF(3)-representability of matroids, J. Combin. Theory Ser. B 32(1982), 112–139.MathSciNetMATHCrossRefGoogle Scholar
  41. Truemper, K.: Partial matroid representations, Europ. J. Combin., 5(1984), 377–394.MathSciNetMATHGoogle Scholar
  42. 1.
    Hassler Whitney, The abstract properties of linear dependence, Amer. J. Math. vol. 57 (1935) pp. 507–533.MathSciNetCrossRefGoogle Scholar
  43. 1.
    C. Kuratowski, Sur le probleme des courbes gauches en Topologie, Fund. Math. vol. 15 (1930) pp. 271–283.MATHGoogle Scholar
  44. 2.
    W. T. Tutte, A class of Abelian groups, Canadian J. Math. vol. 8 (1956) pp. 13–28.MathSciNetMATHCrossRefGoogle Scholar
  45. 3.
    W. T. Tutte, A homotopy theorem for matroids, I. Trans. Amer. Math. Soc. vol. 86 (1958) pp. 144–160.MathSciNetGoogle Scholar
  46. 1.
    R. E. Bixby, l-matrices and a characterization of binary matroids, Discrete Math. 8 (1974), 139–145.MathSciNetMATHCrossRefGoogle Scholar
  47. 2.
    R. E. Bixby, On Reid’s characterization of the matroids, J. Combinatorial Theory Ser. B 26 (1979), 174–204.MathSciNetMATHCrossRefGoogle Scholar
  48. 3.
    F. Harary and D. J. A. Welsh, “Matroids versus Graphs,” Lecture Notes in Mathematics, Vol. 110, pp. 155–170, Springer, Berlin, 1969.Google Scholar
  49. 4.
    A. W. Ingleton, Representation of matroids, in “Combinatorial Mathematics and Its Applications,” pp. 149–167, Academic Press, New York, 1971.Google Scholar
  50. 5.
    A. Lehman, A solution of the Shannon switching game, J. SIAM 12 (1964), 687–725.MathSciNetMATHGoogle Scholar
  51. 6.
    P. D. Seymour, The forbidden minors of binary clutters, J. London Math. Soc. (2), 12 (1976), 356–360.MathSciNetMATHCrossRefGoogle Scholar
  52. 7.
    W. T. Tutte, A homotopy theorem for matroids, I, II, Trans. Amer. Math. Soc. 88 (1958), 144–174.MathSciNetMATHGoogle Scholar
  53. 8.
    W. T. Tutte, Connectivity in matroids, Canad. J. Math. 18 (1966), 1301–1324.MathSciNetMATHCrossRefGoogle Scholar
  54. 9.
    W. T. Tutte, “Introduction to the Theory of Matroids,” RAND Corp. Report R-448-PR, 1966.Google Scholar
  55. 10.
    W. T. Tutte, Lectures on matroids, J. Res. Nat. Bur. Standards Sect. B 69 (1965), 1–47.MathSciNetMATHCrossRefGoogle Scholar
  56. 11.
    D. J. A. Welsh, “Matroid Theory,” Academic Press, London, 1976.MATHGoogle Scholar
  57. 12.
    H. Whitney, On the abstract properties of linear dependence, Amer. J. Math. 57 (1935), 509–533.MathSciNetCrossRefGoogle Scholar
  58. 13.
    T. H. Brylawski and T. D. Lucas, Uniquely representable combinatorial geometries, in “Proc. Internat. Colloq. Combinatorial Theory, Rome, Italy, 1975,” pp. 83–104.Google Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • Saunders Maclane
    • 1
  • W. T. Tutte
    • 2
  • P. D. Seymour
    • 3
  1. 1.Harvard UniversityUSA
  2. 2.University of TorontoTorontoCanada
  3. 3.Department of Pure MathematicsUniversity CollegeSwansea, WalesUK

Personalised recommendations