On a lower bound for the connectivity of the independence complex of a graph

Aharoni, Berger and Ziv proposed a function which is a lower bound for the connectivity of the independence complex of a graph. They conjectured that this bound is optimal for every graph. We give two different arguments which show that the conjecture is false. Given a finite simple graph G, its independence complex IG is defined as the simplicial complex whose vertices are the vertices of G and whose simplices are the independent sets of G. The topology of independence complexes has been studied by a number of authors. In particular, the connectivity of independence complexes has shown to be of interest in the study of Tverberg graphs [5, Theorem 2.2], independent systems of representatives [3, Theorem 2.1] and other important problems. In [3], Aharoni, Berger and Ziv proposed a function ψ defined on graphs which is a lower bound for the connectivity of IG and conjectured that this bound is optimal. No explicit proof of this bound is given in that article, although the corresponding bound for the homological connectivity follows immediately from a result of Meshulam [9, Claim 3.1]. Moreover, a homological version of the conjecture has been considered, as well as reformulations taking into account the existence of counterexamples in which the independence complex is simplyconnected or not [2]. In this note we give an explicit proof of the fact that ψ(G) is a lower bound for the connectivity of IG, we prove that the conjecture is true in the cases where IG is not simplyconnected or where ψ(G) ≤ 1, we show that there exist counterexamples to the conjecture with ψ(G) = 2, and that there are counterexamples in which ψ(G) and the connectivity of IG take arbitrary values l, k with 3 ≤ l < k. The connectivity conn(X) of a topological space X is usually defined as follows: conn(∅) = −2, conn(X) = k if πi(X) = 0 for every 0 ≤ i ≤ k and πk+1(X) 6= 0, and conn(X) = ∞ if πi(X) = 0 for every i ≥ 0. The homological connectivity connH(X) is defined in the same way replacing the homotopy groups πi(X) by the reduced homology groups with integer coefficients H̃i(X). In this context, however, in order to keep the notation of [3], we will use the shifted versions η(X) = conn(X) + 2, ηH(X) = connH(X) + 2. Date: June 1, 2011. 2000 Mathematics Subject Classification. 05C69, 55U10.