A Study On The Graph Formulation Of Union Closed Conjecture

The Union Closed Conjecture, posed by Peter Frankl in 1979, is one of the most renowned problems in Combinatorics. Its appeal stems from the simplicity of its statement and the potential complexity of its solution. The conjecture asserts that in any union-closed family of sets, there exists an element that belongs to at least half of the sets in the family. This paper explores the graph-theoretic formulation of the conjecture and establishes connections between the set-based and graph-based formulations. Through this connection, we derive new results and provide proofs for specific classes of graphs. Additionally, by analyzing the distribution of pendant vertices within various graphs, we demonstrate the validity of the conjecture for a broader range of graph classes.