Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T11:26:02.726Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniqueness of an independent transversal of a family of sets†

Published online by Cambridge University Press:  24 October 2008

Richard A. Brualdi
Richard A. Brualdi
Affiliation:
University of Wisconsin, Madison, Wisconsin, 53706 U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

In his book Transversal Theory (4) Mirsky introduces the problem of finding a necessary and sufficient condition in order that a finite family of sets have a unique independent transversal. The purpose of this note is to give one necessary and sufficient condition. More generally, we give a necessary and sufficient condition in order that a common basis of two rank-finite matroids be unique. We then specialize this result to give an answer to the above question.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 3 , November 1972 , pp. 375 - 380
Copyright
Copyright © Cambridge Philosophical Society 1972

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)Brualdi, R. A.Admissible mappings between dependence spaces. Proc. London Math. Soc. (3) 21 (1970), 296–312.CrossRefGoogle Scholar
(2)Crapo, H. and Rota, G. C.Combinatorial Geometries (preliminary edition). M.I.T. Press (Cambridge, Mass. 1970).Google Scholar
(3)Edmonds, J. and Fulkerson, D. R.Transversals and matroid partition. J. Res. Nat. Bur. Standards, Sect. B 69 (1965), 147–153.CrossRefGoogle Scholar
(4)Mirsky, J.Transversal Theory. (London, Academic Press, 1971.)Google Scholar
(5)Rado, R.A theorem on independence relations. Quart. J. Math. Oxford Ser., 13 (1942), 83–89.CrossRefGoogle Scholar
(6)Tutte, W. T.Lectures on Matroids. J. Red. Nat. Bur. Standards, Sect. B 69 (1965), 1–47.Google Scholar
(7)Whitney, H.On the abstract properties of linear dependence. Amer. J. Math. 57 (1935), 509–533.CrossRefGoogle Scholar