Kalai–Meshulam Euler characteristic conjecture for independence complexes

Let $G$ be a graph. Its independence complex $\mathrm{Ind}(G)$ is the simplicial complex whose faces are the independent sets of $G$, and let $\tilde{\chi}$ denote reduced Euler characteristic. For every induced subgraph $H$ of $G$, consider $\tilde{\chi}(\mathrm{Ind}(H))$.

Kalai–Meshulam Euler characteristic conjecture. For any graph $G$, we have

$$-1 \leq \tilde{\chi}(\mathrm{Ind}(H)) \leq 1$$

for every induced subgraph $H$ of $G$ if and only if there are no induced cycles of length divisible by three in $G$.

The forward and reverse characterization was proved by Chudnovsky, Scott, Seymour, and Spirkl, building on Gauthier's earlier result for graphs with no cycles of length divisible by three, induced or otherwise.