On the topology of simplicial complexes related to 3-connected and Hamiltonian graphs

Using techniques from Robin Formanʹs discrete Morse theory, we obtain information about the homology and homotopy type of some graph complexes. Specifically, we prove that the simplicial complex Δn3 of not 3-connected graphs on n vertices is homotopy equivalent to a wedge of (n−3)·(n−2)!/2 spheres of dimension 2n−4, thereby verifying a conjecture by Babson, Björner, Linusson, Shareshian, and Welker. We also determine a basis for the corresponding nonzero homology group in the CW complex of 3-connected graphs. In addition, we show that the complex Γn of non-Hamiltonian graphs on n vertices is homotopy equivalent to a wedge of two complexes, one of the complexes being the complex Δn2 of not 2-connected graphs on n vertices. The homotopy type of Δn2 has been determined, independently, by the five authors listed above and by Turchin. While Γn and Δn2 are homotopy equivalent for small values on n, they are nonequivalent for n=10.