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Symmetric Union Closed Families

Nived J M

Abstract

We demonstrate that when a graph exhibits a specific type of symmetry, it satisfies the Union Closed Conjecture(UCC). Additionally, we show that certain graph classes, such as Cylindrical Grid Graphs and Torus Grid Graphs also satisfy the conjecture. We prove the known result that the union closed family generated by cyclic translates of a fixed set satisfies the UCC, offering a simpler proof via symmetry arguments. Later, we show that the union closed family generated by the family obtained through cyclically shifting elements from selected translates also satisfies the conjecture.

Symmetric Union Closed Families

Abstract

We demonstrate that when a graph exhibits a specific type of symmetry, it satisfies the Union Closed Conjecture(UCC). Additionally, we show that certain graph classes, such as Cylindrical Grid Graphs and Torus Grid Graphs also satisfy the conjecture. We prove the known result that the union closed family generated by cyclic translates of a fixed set satisfies the UCC, offering a simpler proof via symmetry arguments. Later, we show that the union closed family generated by the family obtained through cyclically shifting elements from selected translates also satisfies the conjecture.

Paper Structure

This paper contains 4 sections, 8 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Symmetry in UCC
  3. Some Interesting Applications
  4. Acknowledgment

Key Result

Proposition 1.1

me Let $G$ be a bipartite graph with vertex partition $X \cup Y$. A vertex $x \in X$ is rare if and only if it is abundant in $\langle \mathcal{F}^{ X} \rangle$.

Theorems & Definitions (15)

  • Proposition 1.1
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • ...and 5 more