The poset on connected graphs is Sperner

TL;DR

This paper proves that the set of all connected graphs on a fixed vertex set, ordered by edge inclusion, forms a Sperner poset where the largest antichain corresponds to the largest level of graphs with the same number of edges.

Contribution

The paper establishes that the poset of connected graphs ordered by edge inclusion has the Sperner property, a new result in graph poset theory.

Findings

Largest antichain equals the largest level in the poset

The poset is graded with levels by number of edges

Connected graphs form a Sperner family

Abstract

Let $\mathcal{G}$ be the set of all connected graphs on vertex set $[n]$. Define the partial ordering $<$ on $\mathcal{G}$ as follows: for $G,H\in \mathcal{G}$ let $G<H$ if $E(G)\subset E(H)$. The poset $(\mathcal{G},<)$ is graded, each level containing the connected graphs with the same number of edges. We prove that $(\mathcal{G},<)$ has the Sperner property, namely that the largest antichain of $(\mathcal{G},<)$ is equal to its largest sized level.