In his remarkable proof from 1978 of the Kneser conjecture, László Lovász, for the first time, used tools from Algebraic Topology to obtain lower bounds on the chromatic number of a graph. For this, he associated certain simplicial (respectively cell) complexes to a graph and then exploited topological invariants of the resulting spaces.

In this project we focus on the geometric, topological, and combinatorial properties of certain interesting classes of coloring complexes.