Abstract:
In this talk I will introduce the notion of star cluster of a simplex in a simplicial complex. We will see that these subcomplexes are particulary useful to study independence complexes of graphs. We use star clusters to establish a question on the homotopy types of independence complexes of triangle-free graphs motivated by work of Engström and Jonsson. We also investigate the connectivity of matching complexes and independence complexes of claw-free graphs, study the relationship between chromatic number and the Lusternik-Schnirelmann category, and give some elementary proofs of known results.