Complexes of graphs with bounded independence number

Abstract

Let G = (V, E) be a graph and n a positive integer. Let In(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 In(G) for various classes of graphs, focusing on the class of graphs with maximum degree bounded by Δ. As an application, we obtain the following result Let G be a claw-free graph with maximum degree at most Δ. Then, every collection of ⌊(Δ2+1)(n−1)⌋+1 independent sets in G has a rainbow independent set of size n. © 2022, The Hebrew University of Jerusalem.