Complexes of graphs with bounded independence number

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