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An extremal problem for non-separable matroids

Part of the Lecture Notes in Mathematics book series (LNM,volume 211)

Abstract

Let E be a finite set,

a matroid on

.

is non-separable if every pair of elements of E are in a common circuit (minimal member of

. The following two theorems are proved:

  1. 1.

    Every nonseparable matroid of rank r and cardinality n must have at least r(n-r) + 1 bases (maximal members of

    ). There is for each r and n, a unique matroid with precisely this many bases.

  2. 2.

    If

    is a nonseparable matroid of rank r on n elements containing more than r(n-r) + 1 bases, then

    must contain at least 2(r−1)(n−r−1) + 1 + r bases if 2r ≤ n and (2r−1)(n−r−1) + 2 bases if 2r≧n.

The research for this paper was conducted while the author was a Visiting Post Graduate Student at the University of Sheffield, Sheffield, England.

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Bibliography

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© 1971 Springer-Verlag

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Dinolt, G.W. (1971). An extremal problem for non-separable matroids. In: Théorie des Matroïdes. Lecture Notes in Mathematics, vol 211. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0061073

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  • DOI: https://doi.org/10.1007/BFb0061073

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-05545-7

  • Online ISBN: 978-3-540-36877-9

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