Fair representation by independent sets

TL;DR

This paper investigates conditions under which hypergraph edges can represent partitions fairly or almost fairly, focusing on specific classes like paths and bipartite matchings, using topological proof methods.

Contribution

It establishes that in certain hypergraph classes, any partition can be almost fairly represented by an edge, and conjectures this for matchings in complete bipartite graphs.

Findings

Paths allow almost fair representation of any partition.

Partitions of $E(K_{n,n})$ into three sets can be almost fairly represented.

Uses topological methods for proofs.

Abstract

For a hypergraph $H$ let $\beta(H)$ 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 $|S\cap V_i|\ge \lfloor\frac{|V_i|}{\beta(H)}\rfloor$ (resp. $|S\cap V_i|\ge \lfloor\frac{|V_i|}{\beta(H)}\rfloor-1$) 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.