Complexes of graphs with bounded independence number

Let $G=(V,E)$ be a graph and $n$ a positive integer. Let $I_n(G)$ be the abstract simplicial complex whose simplices are the subsets of $V$ that do not contain an independent set of size $n$ in $G$. We study the collapsibility numbers of the complexes $I_n(G)$ for various classes of graphs, focusing on the class of graphs with maximum degree bounded by $Delta$. As an application, we obtain the following result: Let $G$ be a claw-free graph with maximum degree at most $Delta$. Then, every collection of $leftlfloorleft(frac{Delta}{2}+1right)(n-1)rightrfloor+1$ independent sets in $G$ has a rainbow independent set of size $n$.