  Chudnovsky, Scott, Seymour and Spirkl recently proved a conjecture by Kalai
and Meshulam stating that the reduced Euler characteristic of the independence
complex of a graph without induced cycles of length divisible by three is in
{-1,0,1}. Gauthier had earlier proved that assuming no cycles of those lengths,
induced or not. Kalai and Meshulam also stated a stronger topological
conjecture, that the total betti numbers are in {0,1}. Towards that we prove an
even stronger statement in the same setting as Gauthier: The independence
complexes are either contractible or homotopy equivalent to spheres. We
conjecture that it also holds in the general setting.

Combinatorics;Algebraic Topology;Geometric Topology