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arXiv · 1611.03196

Fair representation by independent sets

Ron Aharoni·Noga Alon·Eli Berger·Maria Chudnovsky·Dani Kotlar·Martin Loebl·Ran Ziv
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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 {\em fairly} (resp. {\em almost fairly}) a partition $(V_1,V_2, \ldots, V_m)$ of $V(H)$ if $|S\cap V_i|\ge \lfloor\frac{|V_i|}{β(H)}\rfloor$ (resp. $|S\cap V_i|\ge \lfloor\frac{|V_i|}{β(H)}\rfloor-1$) for all $i \le m$. 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.

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Ron Aharoni, Noga Alon, Eli Berger, Maria Chudnovsky, Dani Kotlar, Martin Loebl, Ran Ziv. 2016-11-10. Fair representation by independent sets. https://arxiv.org/abs/1611.03196

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