Given a graph $G$, the /textit{independence complex} $I(G)$ is the simplicial complex whose faces are the independent sets of $V(G)$. Let $b_i$ denote the $i$-th reduced Betti number of $I(G)$, and let $b(G)$ denote the sum of $b_i(G)$'s. A graph is ternary if it does not contain induced cycles with length divisible by three. G. Kalai and K. Meshulam conjectured that $b(G)=2$ and $b(H)/in /{0,1/}$ for every induced subgraph $H$ of $G$ if and only if $G$ is a cycle with length divisible by three. 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.

This is joint work with a graduate student Wentao Zhang in Fudan University.

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