Criterion for acyclicity of flag complexes

Question

Let $\Delta$ be a flag complex on $n$ vertices. Let $r$ be the smallest size of the facets of $\Delta$. Suppose that $2r>n$. Must $\Delta$ be acyclic?

Answer 1

It looks to me like you can prove the stronger property of contractibility by induction, as follows. Let $\Delta$ be the independence complex of graph $G$, as guaranteed by the flag property.

If $\Delta$ is a cone, then $\Delta$ is contractible. This will be the base case, along with dimension 0 (where $\Delta$ has a single point) and dimension 1 (where $\Delta$ is a 1-simplex).

Otherwise, every vertex $v$ has degree at least 1 in $G$. Consider a fixed pair of vertices $v,w$ that are adjacent in $G$. We'll use that the link of $v$ in $\Delta$ is the independence complex of $G \setminus N[v]$ (and similarly for $w$). Now $\operatorname{link}_\Delta v$, $\operatorname{link}_\Delta w$ are contained in the independence complex of $G \setminus \{v,w\}$.

Since we remove at most one vertex from each maximal face in each case, $r$ goes down by at most one in each considered subcomplex, while the number of vertices goes down by at least 2. So by induction, $\operatorname{link}_\Delta v$, $\operatorname{link}_\Delta w$, and the independence complex of $G \setminus {v,w}$ are all contractible.

Now $\Delta$ is the union of the faces that contain $v$, those that contain $w$, and those that contain neither. So $\Delta$ is the union of the subcomplexes $\Delta_1 = v*\operatorname{link} v$, $\Delta_2 = w*\operatorname{link} w$, and $\Delta_0$ the independence complex of $G \setminus \{v,w\}$. It now follows by e.g. Lemma 10.4(ii) of Björner's Topological methods that $\Delta$ is contractible.

I didn't immediately see the answer to the question of whether there's a sensible generalization of some sort to non-flag complexes. The flag property is used above in where $\operatorname{link}_\Delta v$ is the independence complex of $G \setminus N[v]$; also in finding a pair of vertices that are in no common face.

(Updated over initial version to fix a problem with the facet sizes in the induction, in response to a comment of and off-MO discussion with @Hailong Dao.)