The Betti Number of the Independence Complex of Ternary Graphs

Given a graph $G$, the $\mathit{\text{independence}}\mathit{\text{complex}}$ $I(G)$ is the simplicial complex whose faces are the independent sets of $V(G)$. Let ${\tilde{b}}_i$ denote the $i$-th reduced Betti number of $I(G)$, and let $b(G)$ denote the sum of ${\tilde{b}}_i(G)$'s. A graph is ternary if it does not contain induced cycles with length divisible by three. G. Kalai and K. Meshulam conjectured that $b(G)\le 1$ whenever $G$ is ternary. We prove this conjecture. This extends a recent results proved by Chudnovsky, Scott, Seymour and Spirkl that for any ternary graph $G$, the number of independent sets with even cardinality and the independent sets with odd cardinality differ by at most 1.