Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T13:23:14.818Z Has data issue: false hasContentIssue false
  • English
  • Français

On the hyperplanes of a matroid

Published online by Cambridge University Press:  24 October 2008

D. J. A. Welsh
D. J. A. Welsh
Affiliation:
Merton College, Oxford
Get access Rights & Permissions [Opens in a new window]

Extract

Matroid theory was first studied by Whitney (1) as an abstract theory of linear independence in vector spaces. Recently its importance in graph theory has been noticed by Tutte (2), Edmonds (3) and Nash-Williams (4,5). Less interest has been shown in the extremely close relationship between matroids and incidence geometries. In this note we develop the more geometrical aspects of matroid theory, paying particular attention to the fundamental role of the hyperplanes of a matroid in this theory.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 65 , Issue 1 , January 1969 , pp. 11 - 18
Copyright
Copyright © Cambridge Philosophical Society 1969

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Whitney, H.On the abstract properties of linear dependence. Amer. J. Math. 57 (1935), 509–533.CrossRefGoogle Scholar
(2)Tutte, W. T.Lectures on matroids. J. Res. Nat. Bur. Standards Sect. B 69 (1965), 1–47.Google Scholar
(3)Edmonds, J.Minimum partition of a matroid into independent subsets. J. Res. Nat. Bur. Standards Sect. B 69 (1965), 67–72.CrossRefGoogle Scholar
(4)Nash-Williams, C. ST. J. A.Seminar on matroids (Oxford University, 1966).Google Scholar
(5)Nash-Williams, C.ST. J. A.Edge-disjoint spanning trees of finite graphs. J. London Math. Soc. 36 (1961), 445–450.CrossRefGoogle Scholar
(6)Biggs, D. A.Maps of geometries. J. London Math. Soc. 41 (1966), 612–619.Google Scholar
(7)Haer, R.Linear algebra and projective geometry (Acad. Press: New York, 1952).Google Scholar
(8)Maclane, S.Some interpretations of abstract linear dependence in terms of projective geometry. Amer. J. Math. 58 (1936), 236–240.CrossRefGoogle Scholar
(9)Birkhoff, G.Lattice theory (Amer. Math. Soc. Coll. Publ. 25, 236–240). (New York, 1948.)Google Scholar
(10)Birkhoff, G.Abstract linear dependence and lattice. Amer. J. Math. 57 (1935), 800–804.CrossRefGoogle Scholar
(11)Minty, G. J.On the axiomatic foundations of the theories of directed linear graphs, electrical networks and network programming. J. Math. Mech. 15 (1966), 485–520.Google Scholar
(12)Maclane, S.A combinatorial condition for planar graphs. Fund. Math. 28 (1937), 22–32.CrossRefGoogle Scholar
(13)Ore, O. Theory of Graphs (Amer. Math. Soc. Coll. Publ. 38). (1962).Google Scholar