Topological connectedness and independent sets in graphs

An abstract simplicial complex C is said to be k-connected if for each -1 <= d <= k and each continuous map f from the sphere S-d to parallel to C parallel to (the body of the geometric realization of C), the map f can be extended to a continuous map from the ball Bd+1 to parallel to C parallel to. In 2000 a link was discovered between the topological connectedness of the independence complex of a graph and various other important graph parameters to do with colouring and partitioning. When the graph represents some other combinatorial structure, for example when it is the line graph of a hypergraph H, this link can be exploited to obtain information such as lower bounds on the matching number of H. Since its discovery there have been many other applications of this phenomenon to combinatorial problems. The aim of this article is to outline this general method and to describe some of its applications.