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.