A Study On The Graph Formulation Of Union Closed Conjecture

Abstract: The Union Closed 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.