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

Abstract

Using techniques from Robin Forman's discrete Morse theory, we obtain information about the homology and homotopy type of some graph complexes. Specifically, we prove that the simplicial complex Δ n 3 of not 3-connected graphs on n vertices is homotopy equivalent to a wedge of ( n−3)·( n−2)!/2 spheres of dimension 2 n−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 Γ n of non-Hamiltonian graphs on n vertices is homotopy equivalent to a wedge of two complexes, one of the complexes being the complex Δ n 2 of not 2-connected graphs on n vertices. The homotopy type of Δ n 2 has been determined, independently, by the five authors listed above and by Turchin. While Γ n and Δ n 2 are homotopy equivalent for small values on n, they are nonequivalent for n=10.