Graph generated union-closed families of sets

TL;DR

This paper investigates a special case of the union-closed sets conjecture using graph theory, proving a strong version for families generated by graphs and analyzing their local properties.

Contribution

It presents a proof for a strong version of the union-closed sets conjecture in the context of graph-generated families, utilizing local property analysis and Kleitman's lemma.

Findings

Existence of an edge e with limited participation in the family

Proof of a strong version of the union-closed sets conjecture for graph-generated families

Bounds on the proportion of sets containing a given element

Abstract

Let G be a graph with vertices V and edges E. Let F be the union-closed family of sets generated by E. Then F is the family of subsets of V without isolated points. Theorem: There is an edge e belongs to E such that |{U belongs to F | e belongs to U}| =< 1/2|F|. This is equivalent to the following assertion: If H is a union-closed family generated by a family of sets of maximum degree two, then there is an $x$ such that |{U belongs to H | x belongs to U}| > 1/2|H|. This is a special case of the union-closed sets conjecture. To put this result in perspective, a brief overview of research on the union-closed sets conjecture is given. A proof of a strong version of the theorem on graph-generated families of sets is presented. This proof depends on an analysis of the local properties of F and an application of Kleitman's lemma. Much of the proof applies to arbitrary union-closed families and can be used to obtain bounds on |{U belongs to F | e belongs to U}|/|F|.