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On Ideal Minimally Non-packing Clutters

  • Kenji Kashiwabara
  • Tadashi SakumaEmail author
Chapter
Part of the Indian Statistical Institute Series book series (INSIS)

Abstract

We give a scheme to attack the following conjecture proposed by Cornuéjols, Guenin and Margot: “Every ideal minimally non-packing clutter has a transversal of size 2.” The tilde clutter \(\tilde{\mathscr {C}}\) of a clutter \(\mathscr {C}\) is the set of hyperedges of \(\mathscr {C}\) which intersect any minimum transversal in exactly one element. We divide the (non-)existence problem for the conjecture into the following two steps. The first is to construct a candidate (i.e., precore clutter) for a clutter satisfying some proper set of necessary conditions to be a tilde clutter of an ideal minimally non-packing clutter. The second is to construct an ideal minimally non-packing clutter whose tilde clutter is exactly the candidate of the first step. Concerning the first, we give the necessary conditions and reveal the rich structural properties of precore clutters. We also give several necessary conditions and useful tools for the second. Lastly, we demonstrate our scheme on the case of a special clutter, namely, the combinatorial affine planes. That is, we show that a combinatorial affine plane whose blocking number is at least 3cannot be a tilde clutter of an ideal minimally non-packing clutter.

Notes

Acknowledgements

The second author’s research is supported by Grant-in-Aid for Scientific Research (C) (26400185).

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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of General Systems StudiesUniversity of TokyoMeguro-ku, TokyoJapan
  2. 2.Faculty of ScienceYamagata UniversityYamagataJapan

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