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A Remark on Transversal Numbers

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The Mathematics of Paul Erdös II

Part of the book series: Algorithms and Combinatorics ((AC,volume 14))

Abstract

In his classical monograph published in 1935, Dénes König [K] included one of Paul Erdös’s first remarkable results: an infinite version of the Menger theorem. This result (as well as the König-Hall theorem for bipartite graphs, and many related results covered in the book) can be reformulated as a statement about transversals of certain hypergraphs.Let H be a hypergraph with vertex set V(H) and edge set E(H). A subset T C V(H) is called a transversal of H if it meets every edge E E E(H). The transversal number, r(iJ), is defined as the minimum cardinality of a transversal of H. Clearly, r(H) > v(H), where v(H) denotes the maximum number of pairwise disjoint edges of H. In the above mentioned examples, r(H) = v(H) holds for the corresponding hypergraphs. However, in general it is impossible to bound r from above by any function of u, without putting some restriction on the structure of H.

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Author information

Authors and Affiliations

  1. Hungarian Academy of Sciences, Mathematical Institute, Budapest Pf. 127, 1364, Hungary

    János. Pach

Authors
  1. János. Pach
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Editor information

Editors and Affiliations

  1. AT&T Bell Laboratories, 600 Mountain Avenue, 07974, Murray Hill, NJ, USA

    Ronald L. Graham

  2. Department of Applied Mathematics, Charles University, Malostranske nam. 25, 11800, Praha, Czech Republic

    Jaroslav Nešetřil

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© 1997 Springer-Verlag Berlin Heidelberg

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Pach, J. (1997). A Remark on Transversal Numbers. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös II. Algorithms and Combinatorics, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60406-5_28

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  • DOI: https://doi.org/10.1007/978-3-642-60406-5_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64393-4

  • Online ISBN: 978-3-642-60406-5

  • eBook Packages: Springer Book Archive

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