The graph formulation of the union-closed sets conjecture

TL;DR

This paper reformulates Frankl's union-closed sets conjecture as a graph problem, establishing equivalences and proving it for several classes of bipartite graphs, thus advancing understanding of the conjecture.

Contribution

The paper introduces a graph formulation of the union-closed sets conjecture and proves it for multiple classes of bipartite graphs, expanding the scope of cases where it holds.

Findings

Equivalent graph formulation of the conjecture

Proven for bipartite series-parallel graphs

Proven for bipartitioned circular interval graphs

Abstract

In 1979 Frankl conjectured that in a finite non-trivial union-closed collection of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph there are two adjacent vertices each belonging to at most half of the maximal stable sets. In this graph formulation other special cases become natural. The conjecture is trivially true for non-bipartite graphs and we show that it holds also for the classes of chordal bipartite graphs, subcubic bipartite graphs, bipartite series-parallel graphs and bipartitioned circular interval graphs.