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Fair Representation by Independent Sets

  • Ron Aharoni
  • Noga Alon
  • Eli Berger
  • Maria Chudnovsky
  • Dani Kotlar
  • Martin Loebl
  • Ran Ziv
Chapter

Abstract

For a hypergraph H let β(H) denote the minimal number of edges from H covering V (H). An edge S of H is said to represent fairly (resp. almost fairly) a partition (V1, V2, , V m ) of V (H) if \(\vert S \cap V _{i}\vert \geqslant \left \lfloor \frac{\vert V _{i}\vert } {\beta (H)} \right \rfloor\) (resp. \(\vert S \cap V _{i}\vert \geqslant \left \lfloor \frac{\vert V _{i}\vert } {\beta (H)} \right \rfloor - 1\)) for all \(i\leqslant m\). In matroids any partition of V (H) can be represented fairly by some independent set. We look for classes of hypergraphs H in which any partition of V (H) can be represented almost fairly by some edge. We show that this is true when H is the set of independent sets in a path, and conjecture that it is true when H is the set of matchings in K n, n . We prove that partitions of E(K n, n ) into three sets can be represented almost fairly. The methods of proofs are topological.

Notes

Acknowledgements

The authors are grateful to Frédéric Meunier for pointing out an inaccuracy in a previous version of the paper.

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Copyright information

© Springer International publishing AG 2017

Authors and Affiliations

  • Ron Aharoni
    • 1
  • Noga Alon
    • 2
  • Eli Berger
    • 3
  • Maria Chudnovsky
    • 4
  • Dani Kotlar
    • 5
  • Martin Loebl
    • 6
  • Ran Ziv
    • 5
  1. 1.Department of MathematicsTechnionHaifaIsrael
  2. 2.Sackler School of Mathematics and Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  3. 3.Department of MathematicsHaifa UniversityHaifaIsrael
  4. 4.Department of MathematicsPrinceton UniversityPrincetonUSA
  5. 5.Department of Computer ScienceTel-Hai CollegeUpper GalileeIsrael
  6. 6.Department of Applied MathematicsCharles UniversityPrahaCzech Republic

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