A Study On The Graph Formulation Of Union Closed Sets Conjecture

TL;DR

This paper explores the Union Closed Sets Conjecture through a graph-theoretic lens, establishing new connections and extending its validity to broader graph classes by analyzing graph decompositions and pendant vertices.

Contribution

It introduces a graph formulation of the conjecture, linking set theory and graph theory, and proves the conjecture for new classes of graphs based on their structural properties.

Findings

Extended the conjecture's validity to broader graph classes

Connected set-theoretic results with graph-theoretic formulations

Analyzed graph decompositions and pendant vertices for conjecture validation

Abstract

The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union closed family of sets, there exists at least one element that appears in at least half of the sets within the family. We establish the graph-theoretic version of certain set-theoretic results by connecting the set-based and graph-based formulations. We then prove a theorem in which we investigate the conjecture for graphs, focusing on their decompositions and the position of certain pendant vertices. As a result, we extend the validity of the conjecture to a broader class of graph structures.