The homotopy type of the independence complex of graphs with no induced cycles of length divisible by 3

Abstract

We prove Engström’s conjecture that the independence complex of graphs with no induced cycle of length divisible by 3 is either contractible or homotopy equivalent to a sphere. Our result strengthens a result by Zhang and Wu, verifying a conjecture of Kalai and Meshulam which states that the total Betti number of the independence complex of such a graph is at most 1. A weaker conjecture was proved earlier by Chudnovsky, Scott, Seymour, and Spirkl, who showed that in such a graph, the number of independent sets of even size minus the number of independent sets of odd size has values 0, 1, or .