Homotopy types of the Hom complexes of graphs

Abstract

The Hom complex Hom(T,G) of graphs is a CW-complex associated to a pair of graphs T and G, considered in the graph coloring problem. It is known that certain homotopy invariants of Hom(T,G) give lower bounds for the chromatic number of G.For a fixed finite graph T, we show that there is no homotopy invariant of Hom(T,G) which gives an upper bound for the chromatic number of G. More precisely, for a non-bipartite graph G, we construct a graph H such that Hom(T,G) and Hom(T,H) are homotopy equivalent but χ(H) is much larger than χ(G). The equivariant homotopy type of Hom(T,G) is also considered.