A matroid generalization of a result of Dirac

Abstract

This paper generalizes a theorem of Dirac for graphs by proving that ifM is a 3-connected matroid, then, for all pairs {a,b} of distinct elements ofM and all cocircuitsC * ofM, there is a circuit that contains {a,b} and meetsC *. It is also shown that, although the converse of this result fails, the specified condition can be used to characterize 3-connected matroids.

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References

  1. [1]

    R. E. Bixby: A simple theorem on 3-connectivity,Linear Algebra Appl.,45 (1982), 123–126.

    Google Scholar 

  2. [2]

    G. A. Dirac: In abstrakten Graphen vorhande vollständige 4-Graphen und ihre Unterteilungen,Math. Nachr.,22 (1960), 61–85.

    Google Scholar 

  3. [3]

    J. G. Oxley:Matroid Theory, Oxford University Press, New York, 1992.

    Google Scholar 

  4. [4]

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

    Google Scholar 

  5. [5]

    W. T. Tutte: connectivity in matroids,Canad. J. Math. 18 (1966), 1301–1324.

    Google Scholar 

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The author's research was partially supported by a grant from the National Security Agency.

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Oxley, J. A matroid generalization of a result of Dirac. Combinatorica 17, 267–273 (1997). https://doi.org/10.1007/BF01200909

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Mathematics Subject Classification (1991)

  • 05B35
  • 05C40