Independence complexes of claw-free graphs

We study the class of independence complexes of claw-free graphs. The main theorem gives good bounds on the connectivity of these complexes, given bounds for a few subcomplexes of the same class. Two applications are presented. Firstly, we show that the independence complex of a claw-free graph with n vertices and maximal degree d is ( c n / d + ε ) -connected, where c = 2 / 3 . This can be compared with the result of Szabó and Tardos that c = 1 / 2 is optimal with no restrictions on the graphs. Secondly, we calculate the connectivity of a family of complexes used in Babson and Kozlov’s proof of the Lovász conjecture.