## Hamiltonian Cycles and Symmetric Chains in Boolean Lattices

- Oct. 31, 2012
- 3:30 p.m.
- LeConte 312

## Abstract

This is a paper by Noah Streib and William T. Trotter.

Let $B(n)$ be the subset lattice of $\{1,2,\dots,n\}$. Sperner's
theorem states that the width of $B(n)$ is equal to the size of its
biggest level. There have been several elegant proofs of this result,
including an approach that shows that $B(n)$ has a symmetric chain
partition. Another famous result concerning $B(n)$ is that its cover
graph is Hamiltonian. Motivated by these ideas and by the Middle Two
Levels conjecture, they consider posets that have the Hamiltonian
Cycle-Symmetric Chain Partition (HC-SCP) property. A poset of width $w$
has this property if its cover graph has a Hamiltonian cycle which
parses into $w$ symmetric chains. They show that the subset lattices
have the HC-SCP property. I will demonstrate this proof and look more at
open questions
in the area.