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 D 3 n of not 3-connected graphs on n vertices is homotopy equivalent to a wedge of ðn À 3Þ Á ðn À 2Þ!=2 spheres of dimension 2n À 4; thereby verifying a conjecture by Babson, Bjo¨rner, 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 G n of non-Hamiltonian graphs on n vertices is homotopy equivalent to a wedge of two complexes, one of the complexes being the complex D 2 n of not 2-connected graphs on n vertices. The homotopy type of D 2 n has been determined, independently, by the five authors listed above and by Turchin. While G n and D 2 n are homotopy equivalent for small values on n; they are nonequivalent for n ¼ 10: r

Morse theory from an algebraic viewpoint

“…The following is a useful corollary whose homotopical counterpart has been used to show that certain classes of graph complexes have the homotopy type of a wedge of spheres; see [BBL:99], [Jon03], and [Sha01].…”

“…to study the homotopy type and homology of graph complexes (for examples, see Babson et al [BBL:99], Jonsson [Jon03] and Shareshian [Sha01]). …”

“…Another, less general, approach to constructing an algebraic version of Forman's theory is due to Jakob Jonsson [Jon03].…”

Discrete Morse theory and localization

“…In this more general context, a critical cell z is one which does not satisfy a < z < µ(a) for any pair µ(a) > a. The cluster lemma [20,Lem 4.1] or [22,Lem 2] implies that given a generalized acyclic matching on a regular CW complex, there exists a traditional acyclic matching (in the sense of Definition 2.10) with the same set of critical cells 9 . In our third and final calculation, we employ a generalized acyclic matching on S and construct the discrete flow category.…”

Computing Optimal Morse Matchings

“…Applications to questions in combinatorial topology and related fields are numerous: e.g., Babson et al [3], Forman [10], Batzies and Welker [4], and Jonsson [20].…”