Fair Representation by Independent Sets

Ron Aharoni (), Noga Alon (), Eli Berger (), Maria Chudnovsky (), Dani Kotlar (), Martin Loebl () and Ran Ziv ()
Additional contact information
Ron Aharoni: Technion, Department of Mathematics
Noga Alon: Tel Aviv University, Sackler School of Mathematics and Blavatnik School of Computer Science
Eli Berger: Haifa University, Department of Mathematics
Maria Chudnovsky: Princeton University, Department of Mathematics
Dani Kotlar: Tel-Hai College, Department of Computer Science
Martin Loebl: Charles University, Department of Applied Mathematics
Ran Ziv: Tel-Hai College, Department of Computer Science

A chapter in A Journey Through Discrete Mathematics, 2017, pp 31-58 from Springer

Abstract: 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 (V 1, V 2, …, V m ) of V (H) if | S ∩ V i | ≥ | V i | β ( H ) $$\vert S \cap V _{i}\vert \geqslant \left \lfloor \frac{\vert V _{i}\vert } {\beta (H)} \right \rfloor$$ (resp. | S ∩ V i | ≥ | V i | β ( H ) − 1 $$\vert S \cap V _{i}\vert \geqslant \left \lfloor \frac{\vert V _{i}\vert } {\beta (H)} \right \rfloor - 1$$ ) for all i ≤ m $$i\leqslant m$$ . 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.

Keywords: Fair Representation; Hypergraph; Matching Complexity; Borsuk-Ulam Theorem; Matroid Intersection Theorem (search for similar items in EconPapers)
Date: 2017
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-44479-6_2

Ordering information: This item can be ordered from
http://www.springer.com/9783319444796

DOI: 10.1007/978-3-319-44479-6_2

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().