A Study On The Graph Formulation Of Union Closed Sets Conjecture

TL;DR

The paper investigates the Union Closed Sets Conjecture (UCC) through a graph-theoretic lens, bridging the set-based and graph-based formulations via incidence graphs. It establishes a precise correspondence between abundant elements and rare vertices, and introduces a 2-layered vertex decomposition framework that yields a product formula for maximal stable sets, enabling UCC verification for new graph classes. The approach provides a unified method to transfer results between the two formulations and suggests paths for extending to more complex decompositions and non-2-layered intersections, potentially advancing toward the general conjecture. Overall, it presents a novel decomposition-based toolkit for validating UCC in broader graph families and outlines promising directions for future research.

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.

Figures (6)

• Figure 1: An illustration of an incidence family together with its corresponding incidence graph.
• Figure 2: Illustration of a 2-layered vertex in a graph, where the vertex labeled 1 is 2-layered.
• Figure 3: Illustration of Lemma \ref{['lem4']} with graph $H$, stable set $[n]$ consisting of all 2-layered vertices, and the subset $\Gamma \subseteq \Theta^{\mathsf{c}}$.
• Figure 4: An illustration of Lemma \ref{['lem7']}, showing the decomposition of graph $G$ into subgraphs $H$ and $I$. The common vertex set $[n]$ is highlighted together with its partition into $\Theta$, $\Theta^{\mathsf{c}}$, $\Gamma \subseteq \Theta^{\mathsf{c}}$, and $\Psi \subseteq \Theta^{\mathsf{c}} \setminus \Gamma$.
• Figure 5: The bipartite graph shown consists of two bipartite classes represented by red and blue vertices. As all the vertices in red bipartite class has a pendant vertex adjacent to it, the graph satisfies the conjecture.

Theorems & Definitions (26)

• Conjecture 1.1: Graph formulation of UCC
• Proposition 2.1
• proof
• Proposition 3.1
• proof
• Theorem 3.2
• Theorem 3.3
• Lemma 3.4
• proof
• Lemma 3.5