Symmetric Union Closed Families

TL;DR

This paper proves that certain symmetric graph classes and union closed families generated by cyclic shifts satisfy the Union Closed Conjecture, providing new proofs and extending known results through symmetry-based arguments.

Contribution

It introduces symmetry-based methods to prove the Union Closed Conjecture for specific graph classes and union closed families, simplifying existing proofs and extending their applicability.

Findings

Cylindrical and Torus Grid Graphs satisfy the UCC

Union closed families generated by cyclic translates satisfy the UCC

Symmetry arguments simplify proofs of the UCC for certain 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.