Star clusters in independence complexes of graphs

JONATHAN ARIEL BARMAK

Oberwolfach

Conferencia; Oberwolfach-Topological and Geometric Combinatorics; 2011

In this talk I will introduce the notion of star cluster of a simplex ina simplicial complex. These subcomplexes are useful to study problemsrelating homotopy invariants of the independence complex IG of a graphG and combinatorial properties of G. In particular I will show how thestrong Lusternik-Schnirelmann category of IG plus one gives a lower bound for the chromatic number of G. I will also use star clusters to investigate connectivity of independence complexes of claw-free graphs and matching complexes, and to classify the homotopy types of independence complexes of triangle-free graphs, answering questions arisen from works of Engström and Jonsson. The analogous problem will be considered for other classes of graphs as well.